Theorem necon3ad 2462

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 2421	. 2 |- ( A =/= B <-> -. A = B )
2		necon3ad.1	. . 3 |- ( ph -> ( ps -> A = B ) )
3	2	con3d 640	. 2 |- ( ph -> ( -. A = B -> -. ps ) )
4	1, 3	biimtrid 152	1 |- ( ph -> ( A =/= B -> -. ps ) )

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-in1 623  ax-in2 624

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

This theorem is used by:  necon3d  2464  disjnim  4120  fodjumkvlemres  7499  nlt1pig  7708  eucalglt  12835  nprm  12901  pcprmpw2  13112  pcmpt  13122  expnprm  13132  0nnei  15254  2sqlem8a  16241  bj-charfunr  16836