Theorem necon3bd 2383

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 626	. 2 |- ( ph -> ( -. ps -> -. A = B ) )
3		df-ne 2341	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	syl6ibr 161	1 |- ( ph -> ( -. ps -> A =/= B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1348   =/= wne 2340

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 609  ax-in2 610

This theorem depends on definitions:  df-bi 116  df-ne 2341

This theorem is referenced by:  nelne1  2430  nelne2  2431  nssne1  3205  nssne2  3206  disjne  3468  difsn  3717  nbrne1  4008  nbrne2  4009  ac6sfi  6876  indpi  7304  zneo  9313  pc2dvds  12283  pcadd  12293  oddprmdvds  12306  lgsne0  13733