Theorem necon2ai 2394

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 622	. 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:  necon2i  2396  neneqad  2419  intexr  4136  iin0r  4155  tfrlemisucaccv  6304  pm54.43  7167  renepnf  7967  renemnf  7968  lt0ne0d  8432  nnne0  8906  nn0nepnf  9206  hashennn  10714  bj-intexr  13943