Theorem necon2ai 2418

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 628	. 2 |- ( ph -> -. A = B )
3		df-ne 2365	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	sylibr 134	1 |- ( ph -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1364   =/= wne 2364

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 615  ax-in2 616

This theorem depends on definitions:  df-bi 117  df-ne 2365

This theorem is referenced by:  necon2i  2420  neneqad  2443  intexr  4180  iin0r  4199  tfrlemisucaccv  6380  pm54.43  7252  renepnf  8069  renemnf  8070  lt0ne0d  8534  nnne0  9012  nn0nepnf  9314  hashennn  10854  bj-intexr  15470