Theorem necon4bid 2583

Description: Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.)

Hypothesis

Ref	Expression
necon4bid.1	|- (ph -> (A =/= B <-> C =/= D))

Assertion

Ref	Expression
necon4bid	|- (ph -> (A = B <-> C = D))

Proof of Theorem necon4bid

Step	Hyp	Ref	Expression
1		necon4bid.1	. . 3 |- (ph -> (A =/= B <-> C =/= D))
2	1	necon2bbid 2575	. 2 |- (ph -> (C = D <-> -. A =/= B))
3		nne 2521	. 2 |- (-. A =/= B <-> A = B)
4	2, 3	syl6rbb 253	1 |- (ph -> (A = B <-> C = D))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   = 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:  nebi  2588