Theorem necon2ai 2259

Description: Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)

Hypothesis

Ref	Expression
necon2ai.1	|- ( A = B -> -. ph )

Assertion

Ref	Expression
necon2ai	|- ( ph -> A =/= B )

Proof of Theorem necon2ai

Step	Hyp	Ref	Expression
1		necon2ai.1	. . 3 |- ( A = B -> -. ph )
2	1	con2i 557	. 2 |- ( ph -> -. A = B )
3		df-ne 2206	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	sylibr 137	1 |- ( ph -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1243   =/= wne 2204

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545

This theorem depends on definitions:  df-bi 110  df-ne 2206

This theorem is referenced by:  necon2i  2261  neneqad  2284  intexr  3904  iin0r  3922  tfrlemisucaccv  5939  renepnf  7073  renemnf  7074  lt0ne0d  7505  nnne0  7942  bj-intexr  10028