Theorem necon3i 2253

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 2249	. 2 |- ( ( A = B -> C = D ) -> ( C =/= D -> A =/= B ) )
4	1, 3	ax-mp 7	1 |- ( C =/= D -> A =/= B )

Syntax hints:   -> wi 4   = wceq 1243   =/= wne 2204

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545

This theorem depends on definitions:  df-bi 110  df-ne 2206

This theorem is referenced by: (None)