Theorem necon3bd 2554

Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

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		nne 2521	. . 3 |- (-. A =/= B <-> A = B)
2		necon3bd.1	. . 3 |- (ph -> (A = B -> ps))
3	1, 2	syl5bi 208	. 2 |- (ph -> (-. A =/= B -> ps))
4	3	con1d 116	1 |- (ph -> (-. ps -> A =/= B))

Syntax hints:  -. wn 3   -> wi 4   = wceq 1642   =/= wne 2517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-ne 2519

This theorem is referenced by:  nelne1  2606  nelne2  2607  nssne1  3328  nssne2  3329  disjne  3597  difsn  3846  nbrne1  4657  nbrne2  4658