Theorem necon3bd 2351

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 620	. 2 |- ( ph -> ( -. ps -> -. A = B ) )
3		df-ne 2309	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	syl6ibr 161	1 |- ( ph -> ( -. ps -> A =/= B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1331   =/= wne 2308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604

This theorem depends on definitions:  df-bi 116  df-ne 2309

This theorem is referenced by:  nelne1  2398  nelne2  2399  nssne1  3155  nssne2  3156  disjne  3416  difsn  3657  nbrne1  3947  nbrne2  3948  ac6sfi  6792  indpi  7150  zneo  9152