Theorem necon2ad 2393

Description: Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)

Hypothesis

Ref	Expression
necon2ad.1	|- ( ph -> ( A = B -> -. ps ) )

Assertion

Ref	Expression
necon2ad	|- ( ph -> ( ps -> A =/= B ) )

Proof of Theorem necon2ad

Step	Hyp	Ref	Expression
1		necon2ad.1	. . 3 |- ( ph -> ( A = B -> -. ps ) )
2	1	con2d 614	. 2 |- ( ph -> ( ps -> -. A = B ) )
3		df-ne 2337	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	syl6ibr 161	1 |- ( ph -> ( ps -> A =/= B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1343   =/= wne 2336

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 604  ax-in2 605

This theorem depends on definitions:  df-bi 116  df-ne 2337

This theorem is referenced by:  necon2d  2395  prneimg  3754  tz7.2  4332  nordeq  4521  pr2ne  7148  ltne  7983  apne  8521  xrltne  9749  npnflt  9751  nmnfgt  9754  ge0nemnf  9760  rpexp  12085  sqrt2irr  12094  pcgcd1  12259  lgsmod  13567