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
@@ -2705,11 +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.
 
+  // 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;
+  }
 
-  // (10 predicates * 2 constants) possible combinations. Cases not handled here
-  // require a 'not' of the LHS, so those must be transformed in InstCombine.
   if (match(RHS, m_Zero())) {
     switch (Pred) {
     case CmpInst::ICMP_NE:  // X !=  0 -> X
@@ -2717,13 +2722,6 @@
     case CmpInst::ICMP_SLT: // X <s  0 -> X
       return LHS;
 
-    case CmpInst::ICMP_EQ:
-      // (X ^ 1) == 0 -> X
-      if (match(LHS, m_Xor(m_Value(X), m_One()))) {
-        return X;
-      }
-      break;
-
     case CmpInst::ICMP_ULT: // X <u  0 -> false
     case CmpInst::ICMP_SGT: // X >s  0 -> false
       return getFalse(ITy);
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
@@ -186,8 +186,64 @@
   ret i1 %cmp
 }
 
-define i1 @not_cmp_equal(i1 %x) {
-; CHECK-LABEL: @not_cmp_equal(
+; 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