Theorem necon3i 2556

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 2555	. 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 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:  addcnnul  4454  tfin11  4494  eventfin  4518  oddtfin  4519  xpnz  5046