Theorem necon2ai 2309

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 592	. 2 |- ( ph -> -. A = B )
3		df-ne 2256	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	sylibr 132	1 |- ( ph -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1289   =/= wne 2255

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580

This theorem depends on definitions:  df-bi 115  df-ne 2256

This theorem is referenced by:  necon2i  2311  neneqad  2334  intexr  3986  iin0r  4004  tfrlemisucaccv  6090  pm54.43  6816  renepnf  7533  renemnf  7534  lt0ne0d  7989  nnne0  8448  nn0nepnf  8742  hashennn  10184  bj-intexr  11754