Theorem necon2ad 2477

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 633	. 2 |- ( ph -> ( ps -> -. A = B ) )
3		df-ne 2421	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	imbitrrdi 162	1 |- ( ph -> ( ps -> A =/= B ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   = wceq 1402   =/= wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624

This proof depends on definitions:  df-bi 117  df-ne 2421

This theorem is used by:  necon2d  2479  prneimg  3899  tz7.2  4499  nordeq  4691  pr2ne  7539  ltne  8411  apne  8954  xrltne  10226  npnflt  10228  nmnfgt  10231  ge0nemnf  10237  rpexp  12951  sqrt2irr  12960  pcgcd1  13130  nzrunit  14579  lgsmod  16311