Theorem necon3bd 2292

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

Hypothesis

Ref	Expression
necon3bd.1	|- ( ph -> ( A = B -> ps ) )

Assertion

Ref	Expression
necon3bd	|- ( ph -> ( -. ps -> A =/= B ) )

Proof of Theorem necon3bd

Step	Hyp	Ref	Expression
1		necon3bd.1	. . 3 |- ( ph -> ( A = B -> ps ) )
2	1	con3d 594	. 2 |- ( ph -> ( -. ps -> -. A = B ) )
3		df-ne 2250	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	syl6ibr 160	1 |- ( ph -> ( -. ps -> A =/= B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1285   =/= wne 2249

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578

This theorem depends on definitions:  df-bi 115  df-ne 2250

This theorem is referenced by:  nelne1  2339  nelne2  2340  nssne1  3066  nssne2  3067  disjne  3318  difsn  3548  nbrne1  3828  nbrne2  3829  ac6sfi  6544  indpi  6804  zneo  8743