diff --git a/libcxx/include/CMakeLists.txt b/libcxx/include/CMakeLists.txt
--- a/libcxx/include/CMakeLists.txt
+++ b/libcxx/include/CMakeLists.txt
@@ -202,14 +202,6 @@
 add_custom_target(cxx-generated-config ALL
   DEPENDS ${LIBCXX_BINARY_DIR}/__generated_config)
 
-# In some build configurations (like bootstrapping clang), we need to be able to
-# install the libcxx headers before the CMake configuration for libcxx runs. Making
-# the name of this target configurable allows LLVM/runtimes/CMakeLists.txt to
-# add this subdirectory to the LLVM build to put libcxx's headers in place
-# before libcxx's build configuration is run.
-if (NOT CXX_HEADER_TARGET)
-  set(CXX_HEADER_TARGET cxx-headers)
-endif()
 if(LIBCXX_HEADER_DIR)
   set(output_dir ${LIBCXX_HEADER_DIR}/include/c++/v1)
 
@@ -234,23 +226,23 @@
   list(APPEND out_files ${dst})
   add_custom_target(generate-cxx-headers DEPENDS ${out_files})
 
-  add_library(${CXX_HEADER_TARGET} INTERFACE)
-  add_dependencies(${CXX_HEADER_TARGET} generate-cxx-headers ${LIBCXX_CXX_ABI_HEADER_TARGET})
+  add_library(cxx-headers INTERFACE)
+  add_dependencies(cxx-headers generate-cxx-headers ${LIBCXX_CXX_ABI_HEADER_TARGET})
   # TODO: Use target_include_directories once we figure out why that breaks the runtimes build
   if("${CMAKE_CXX_COMPILER_ID}" STREQUAL "MSVC" OR "${CMAKE_CXX_SIMULATE_ID}" STREQUAL "MSVC")
-    target_compile_options(${CXX_HEADER_TARGET} INTERFACE /I "${output_dir}")
+    target_compile_options(cxx-headers INTERFACE /I "${output_dir}")
   else()
-    target_compile_options(${CXX_HEADER_TARGET} INTERFACE -I "${output_dir}")
+    target_compile_options(cxx-headers INTERFACE -I "${output_dir}")
   endif()
 
   # Make sure the generated __config_site header is included when we build the library.
   if("${CMAKE_CXX_COMPILER_ID}" STREQUAL "MSVC" OR "${CMAKE_CXX_SIMULATE_ID}" STREQUAL "MSVC")
-    target_compile_options(${CXX_HEADER_TARGET} INTERFACE /FI "${LIBCXX_BINARY_DIR}/__config_site")
+    target_compile_options(cxx-headers INTERFACE /FI "${LIBCXX_BINARY_DIR}/__config_site")
   else()
-    target_compile_options(${CXX_HEADER_TARGET} INTERFACE -include "${LIBCXX_BINARY_DIR}/__config_site")
+    target_compile_options(cxx-headers INTERFACE -include "${LIBCXX_BINARY_DIR}/__config_site")
   endif()
 else()
-  add_library(${CXX_HEADER_TARGET} INTERFACE)
+  add_library(cxx-headers INTERFACE)
 endif()
 
 if (LIBCXX_INSTALL_HEADERS)
@@ -258,7 +250,7 @@
     get_filename_component(dir ${file} DIRECTORY)
     install(FILES ${file}
       DESTINATION ${LIBCXX_INSTALL_HEADER_PREFIX}include/c++/v1/${dir}
-      COMPONENT ${CXX_HEADER_TARGET}
+      COMPONENT cxx-headers
       PERMISSIONS OWNER_READ OWNER_WRITE GROUP_READ WORLD_READ
     )
   endforeach()
@@ -268,15 +260,15 @@
     DESTINATION ${LIBCXX_INSTALL_HEADER_PREFIX}include/c++/v1
     PERMISSIONS OWNER_READ OWNER_WRITE GROUP_READ WORLD_READ
     RENAME __config
-    COMPONENT ${CXX_HEADER_TARGET})
+    COMPONENT cxx-headers)
 
   if (NOT CMAKE_CONFIGURATION_TYPES)
-    add_custom_target(install-${CXX_HEADER_TARGET}
-                      DEPENDS ${CXX_HEADER_TARGET} cxx-generated-config
+    add_custom_target(install-cxx-headers
+                      DEPENDS cxx-headers cxx-generated-config
                       COMMAND "${CMAKE_COMMAND}"
-                              -DCMAKE_INSTALL_COMPONENT=${CXX_HEADER_TARGET}
+                              -DCMAKE_INSTALL_COMPONENT=cxx-headers
                               -P "${CMAKE_BINARY_DIR}/cmake_install.cmake")
     # Stripping is a no-op for headers
-    add_custom_target(install-${CXX_HEADER_TARGET}-stripped DEPENDS install-${CXX_HEADER_TARGET})
+    add_custom_target(install-cxx-headers-stripped DEPENDS install-cxx-headers)
   endif()
 endif()
diff --git a/llvm/runtimes/CMakeLists.txt b/llvm/runtimes/CMakeLists.txt
--- a/llvm/runtimes/CMakeLists.txt
+++ b/llvm/runtimes/CMakeLists.txt
@@ -98,9 +98,17 @@
 
   include(CheckLibraryExists)
   include(CheckCCompilerFlag)
+  include(CMakePushCheckState)
 
-  # We don't have libc++ (yet).
-  set(CMAKE_REQUIRED_FLAGS "${CMAKE_REQUIRED_FLAGS} -nostdlib++")
+  cmake_push_check_state()
+
+  # We don't have libc++ (yet)...
+  set(CMAKE_REQUIRED_FLAGS "${CMAKE_REQUIRED_FLAGS} -nostdinc++ -nostdlib++")
+
+  # ...but we need access to libc++ headers for CMake checks to succeed.
+  if (LLVM_EXTERNAL_LIBCXX_SOURCE_DIR AND "libcxx" IN_LIST LLVM_ENABLE_RUNTIMES)
+    set(CMAKE_REQUIRED_FLAGS "${CMAKE_REQUIRED_FLAGS} -isystem ${LLVM_EXTERNAL_LIBCXX_SOURCE_DIR}/include")
+  endif()
 
   # Avoid checking whether the compiler is working.
   set(LLVM_COMPILER_CHECKED ON)
@@ -110,8 +118,7 @@
   include(HandleLLVMOptions)
   include(FindPythonInterp)
 
-  # Remove the -nostdlib++ option we've added earlier.
-  string(REPLACE "-nostdlib++" "" CMAKE_REQUIRED_FLAGS "${CMAKE_REQUIRED_FLAGS}")
+  cmake_pop_check_state()
 
   # Use libtool instead of ar if you are both on an Apple host, and targeting Apple.
   if(CMAKE_HOST_APPLE AND APPLE)
@@ -215,15 +222,6 @@
 
 else() # if this is included from LLVM's CMake
   include(LLVMExternalProjectUtils)
-  if (LLVM_EXTERNAL_LIBCXX_SOURCE_DIR AND "libcxx" IN_LIST LLVM_ENABLE_RUNTIMES)
-    # This looks wrong, but libcxx's build actually wants the header dir to be
-    # the root build dir, not the include directory.
-    set(LIBCXX_BINARY_DIR ${LLVM_BINARY_DIR})
-    set(LIBCXX_SOURCE_DIR ${LLVM_EXTERNAL_LIBCXX_SOURCE_DIR})
-    set(LIBCXX_HEADER_DIR ${LLVM_BINARY_DIR})
-    set(CXX_HEADER_TARGET runtime-libcxx-headers)
-    add_subdirectory(${LLVM_EXTERNAL_LIBCXX_SOURCE_DIR}/include ${CXX_HEADER_TARGET})
-  endif()
 
   if(NOT LLVM_BUILD_RUNTIMES)
     set(EXTRA_ARGS EXCLUDE_FROM_ALL)
@@ -414,7 +412,7 @@
 
     llvm_ExternalProject_Add(runtimes
                              ${CMAKE_CURRENT_SOURCE_DIR}
-                             DEPENDS ${ARG_DEPENDS} ${CXX_HEADER_TARGET}
+                             DEPENDS ${ARG_DEPENDS}
                              # Builtins were built separately above
                              CMAKE_ARGS -DCOMPILER_RT_BUILD_BUILTINS=Off
                                         -DLLVM_INCLUDE_TESTS=${LLVM_INCLUDE_TESTS}
@@ -520,7 +518,7 @@
 
     llvm_ExternalProject_Add(runtimes-${name}
                              ${CMAKE_CURRENT_SOURCE_DIR}
-                             DEPENDS ${${name}_deps} ${CXX_HEADER_TARGET}
+                             DEPENDS ${${name}_deps}
                              # Builtins were built separately above
                              CMAKE_ARGS -DCOMPILER_RT_BUILD_BUILTINS=Off
                                         -DLLVM_INCLUDE_TESTS=${LLVM_INCLUDE_TESTS}