Theorem necon2ad 2457

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 627	. 2 |- ( ph -> ( ps -> -. A = B ) )
3		df-ne 2401	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	imbitrrdi 162	1 |- ( ph -> ( ps -> A =/= B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1395   =/= wne 2400

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 617  ax-in2 618

This theorem depends on definitions:  df-bi 117  df-ne 2401

This theorem is referenced by:  necon2d  2459  prneimg  3851  tz7.2  4444  nordeq  4635  pr2ne  7361  ltne  8227  apne  8766  xrltne  10005  npnflt  10007  nmnfgt  10010  ge0nemnf  10016  rpexp  12670  sqrt2irr  12679  pcgcd1  12846  nzrunit  14146  lgsmod  15699