Theorem necon3i 2384

Description: Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.)

Hypothesis

Ref	Expression
necon3i.1	|- ( A = B -> C = D )

Assertion

Ref	Expression
necon3i	|- ( C =/= D -> A =/= B )

Proof of Theorem necon3i

Step	Hyp	Ref	Expression
1		necon3i.1	. 2 |- ( A = B -> C = D )
2		id 19	. . 3 |- ( ( A = B -> C = D ) -> ( A = B -> C = D ) )
3	2	necon3d 2380	. 2 |- ( ( A = B -> C = D ) -> ( C =/= D -> A =/= B ) )
4	1, 3	ax-mp 5	1 |- ( C =/= D -> A =/= B )

Syntax hints:   -> wi 4   = wceq 1343   =/= wne 2336

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 604  ax-in2 605

This theorem depends on definitions:  df-bi 116  df-ne 2337

This theorem is referenced by:  expnprm  12283  lgsne0  13579