Theorem necon3ad 2382

Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

Hypothesis

Ref	Expression
necon3ad.1	|- ( ph -> ( ps -> A = B ) )

Assertion

Ref	Expression
necon3ad	|- ( ph -> ( A =/= B -> -. ps ) )

Proof of Theorem necon3ad

Step	Hyp	Ref	Expression
1		df-ne 2341	. 2 |- ( A =/= B <-> -. A = B )
2		necon3ad.1	. . 3 |- ( ph -> ( ps -> A = B ) )
3	2	con3d 626	. 2 |- ( ph -> ( -. A = B -> -. ps ) )
4	1, 3	syl5bi 151	1 |- ( ph -> ( A =/= B -> -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1348   =/= wne 2340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 609  ax-in2 610

This theorem depends on definitions:  df-bi 116  df-ne 2341

This theorem is referenced by:  necon3d  2384  disjnim  3980  fodjumkvlemres  7135  nlt1pig  7303  eucalglt  12011  nprm  12077  pcprmpw2  12286  pcmpt  12295  expnprm  12305  0nnei  12947  2sqlem8a  13752  bj-charfunr  13845