diff --git a/llvm/lib/Analysis/InstructionSimplify.cpp b/llvm/lib/Analysis/InstructionSimplify.cpp
--- a/llvm/lib/Analysis/InstructionSimplify.cpp
+++ b/llvm/lib/Analysis/InstructionSimplify.cpp
@@ -2705,8 +2705,16 @@
     return nullptr;
 
   // A boolean compared to true/false can be simplified in 14 out of the 20
-  // (10 predicates * 2 constants) possible combinations. Cases not handled here
-  // require a 'not' of the LHS, so those must be transformed in InstCombine.
+  // (10 predicates * 2 constants) possible combinations.
+
+  // If LHS is of the form not(X), inverse the predicate.
+  // E.g not(X) == 0 -> X != 0
+  Value *X;
+  if (match(LHS, m_Xor(m_Value(X), m_One()))) {
+    Pred = CmpInst::getInversePredicate(Pred);
+    LHS = X;
+  }
+
   if (match(RHS, m_Zero())) {
     switch (Pred) {
     case CmpInst::ICMP_NE:  // X !=  0 -> X
diff --git a/llvm/test/Transforms/InstSimplify/icmp.ll b/llvm/test/Transforms/InstSimplify/icmp.ll
--- a/llvm/test/Transforms/InstSimplify/icmp.ll
+++ b/llvm/test/Transforms/InstSimplify/icmp.ll
@@ -186,6 +186,72 @@
   ret i1 %cmp
 }
 
+; Test simplification with not in LHS
+
+define i1 @not_cmp_eq(i1 %x) {
+; CHECK-LABEL: @not_cmp_eq(
+; CHECK-NEXT:    ret i1 %x
+;
+  %not = xor i1 %x, true
+  %cmp = icmp eq i1 %not, 0
+  ret i1 %cmp
+}
+
+define i1 @not_cmp_ule(i1 %x) {
+; CHECK-LABEL: @not_cmp_ule(
+; CHECK-NEXT:    ret i1 %x
+;
+  %not = xor i1 %x, true
+  %cmp = icmp ule i1 %not, 0
+  ret i1 %cmp
+}
+
+define i1 @not_cmp_sge(i1 %x) {
+; CHECK-LABEL: @not_cmp_sge(
+; CHECK-NEXT:    ret i1 %x
+;
+  %not = xor i1 %x, true
+  %cmp = icmp sge i1 %not, 0
+  ret i1 %cmp
+}
+
+define i1 @not_cmp_ne(i1 %x) {
+; CHECK-LABEL: @not_cmp_ne(
+; CHECK-NEXT:    ret i1 %x
+;
+  %not = xor i1 %x, true
+  %cmp = icmp ne i1 %not, true
+  ret i1 %cmp
+}
+
+define i1 @not_cmp_ult(i1 %x) {
+; CHECK-LABEL: @not_cmp_ult(
+; CHECK-NEXT:    ret i1 %x
+;
+  %not = xor i1 %x, true
+  %cmp = icmp ult i1 %not, 1
+  ret i1 %cmp
+}
+
+define i1 @not_cmp_sgt(i1 %x) {
+; CHECK-LABEL: @not_cmp_sgt(
+; CHECK-NEXT:    ret i1 %x
+;
+  %not = xor i1 %x, true
+  %cmp = icmp sgt i1 %not, -1
+  ret i1 %cmp
+}
+
+define i1 @not_cmp_sext(i1 %x) {
+; CHECK-LABEL: @not_cmp_sext(
+; CHECK-NEXT:    ret i1 %x
+;
+  %not = xor i1 %x, true
+  %t0 = sext i1 %not to i32
+  %cmp = icmp eq i32 %t0, 0
+  ret i1 %cmp
+}
+
 ; Don't crash matching recurrences/invertible ops.
 
 define void @PR50191(i32 %x) {