Theorem necon2ai 2456

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 632	. 2 |- ( ph -> -. A = B )
3		df-ne 2403	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	sylibr 134	1 |- ( ph -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1397   =/= wne 2402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620

This theorem depends on definitions:  df-bi 117  df-ne 2403

This theorem is referenced by:  necon2i  2458  neneqad  2481  intexr  4240  iin0r  4259  tfrlemisucaccv  6490  pm54.43  7394  renepnf  8226  renemnf  8227  lt0ne0d  8692  nnne0  9170  nn0nepnf  9472  hashennn  11041  bj-intexr  16503