Theorem necon2d 2365

Description: Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)

Hypothesis

Ref	Expression
necon2d.1	|- ( ph -> ( A = B -> C =/= D ) )

Assertion

Ref	Expression
necon2d	|- ( ph -> ( C = D -> A =/= B ) )

Proof of Theorem necon2d

Step	Hyp	Ref	Expression
1		necon2d.1	. . 3 |- ( ph -> ( A = B -> C =/= D ) )
2		df-ne 2307	. . 3 |- ( C =/= D <-> -. C = D )
3	1, 2	syl6ib 160	. 2 |- ( ph -> ( A = B -> -. C = D ) )
4	3	necon2ad 2363	1 |- ( ph -> ( C = D -> A =/= B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1331   =/= wne 2306

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 2307

This theorem is referenced by:  map0g  6575  hashprg  10547