Theorem necon1d 2586

Description: Contrapositive law deduction for inequality. (Contributed by NM, 28-Dec-2008.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon1d.1	|- (ph -> (A =/= B -> C = D))

Assertion

Ref	Expression
necon1d	|- (ph -> (C =/= D -> A = B))

Proof of Theorem necon1d

Step	Hyp	Ref	Expression
1		necon1d.1	. . 3 |- (ph -> (A =/= B -> C = D))
2		nne 2521	. . 3 |- (-. C =/= D <-> C = D)
3	1, 2	syl6ibr 218	. 2 |- (ph -> (A =/= B -> -. C =/= D))
4	3	necon4ad 2578	1 |- (ph -> (C =/= D -> 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: (None)