Theorem necon3ad 2445

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 2404	. 2 |- ( A =/= B <-> -. A = B )
2		necon3ad.1	. . 3 |- ( ph -> ( ps -> A = B ) )
3	2	con3d 636	. 2 |- ( ph -> ( -. A = B -> -. ps ) )
4	1, 3	biimtrid 152	1 |- ( ph -> ( A =/= B -> -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1398   =/= wne 2403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 619  ax-in2 620

This theorem depends on definitions:  df-bi 117  df-ne 2404

This theorem is referenced by:  necon3d  2447  disjnim  4083  fodjumkvlemres  7401  nlt1pig  7604  eucalglt  12692  nprm  12758  pcprmpw2  12969  pcmpt  12979  expnprm  12989  0nnei  14947  2sqlem8a  15924  bj-charfunr  16509