Theorem necon2ad 2459

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 629	. 2 |- ( ph -> ( ps -> -. A = B ) )
3		df-ne 2403	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	imbitrrdi 162	1 |- ( ph -> ( ps -> A =/= B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1397   =/= wne 2402

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 619  ax-in2 620

This theorem depends on definitions:  df-bi 117  df-ne 2403

This theorem is referenced by:  necon2d  2461  prneimg  3857  tz7.2  4451  nordeq  4642  pr2ne  7396  ltne  8263  apne  8802  xrltne  10047  npnflt  10049  nmnfgt  10052  ge0nemnf  10058  rpexp  12724  sqrt2irr  12733  pcgcd1  12900  nzrunit  14201  lgsmod  15754