Theorem necon3bi 2390

Description: Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

Hypothesis

Ref	Expression
necon3bi.1	|- ( A = B -> ph )

Assertion

Ref	Expression
necon3bi	|- ( -. ph -> A =/= B )

Proof of Theorem necon3bi

Step	Hyp	Ref	Expression
1		necon3bi.1	. . 3 |- ( A = B -> ph )
2	1	con3i 627	. 2 |- ( -. ph -> -. A = B )
3		df-ne 2341	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	sylibr 133	1 |- ( -. ph -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1348   =/= wne 2340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610

This theorem depends on definitions:  df-bi 116  df-ne 2341

This theorem is referenced by:  pwne  4146  sucpw1ne3  7209  nltpnft  9771  ngtmnft  9774