Theorem necon3ad 2378

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 2337	. 2 |- ( A =/= B <-> -. A = B )
2		necon3ad.1	. . 3 |- ( ph -> ( ps -> A = B ) )
3	2	con3d 621	. 2 |- ( ph -> ( -. A = B -> -. ps ) )
4	1, 3	syl5bi 151	1 |- ( ph -> ( A =/= B -> -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1343   =/= wne 2336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 604  ax-in2 605

This theorem depends on definitions:  df-bi 116  df-ne 2337

This theorem is referenced by:  necon3d  2380  disjnim  3973  fodjumkvlemres  7123  nlt1pig  7282  eucalglt  11989  nprm  12055  pcprmpw2  12264  pcmpt  12273  expnprm  12283  0nnei  12803  2sqlem8a  13608  bj-charfunr  13702