Theorem necon2ai 2299

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 589	. 2 |- ( ph -> -. A = B )
3		df-ne 2246	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	sylibr 132	1 |- ( ph -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1284   =/= wne 2245

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 576  ax-in2 577

This theorem depends on definitions:  df-bi 115  df-ne 2246

This theorem is referenced by:  necon2i  2301  neneqad  2324  intexr  3925  iin0r  3943  tfrlemisucaccv  5962  pm54.43  6459  renepnf  7166  renemnf  7167  lt0ne0d  7614  nnne0  8067  bj-intexr  10699