Theorem nebi 2588

Description: Contraposition law for inequality. (Contributed by NM, 28-Dec-2008.)

Assertion

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

Proof of Theorem nebi

Step	Hyp	Ref	Expression
1		id 19	. . 3 |- ((A = B <-> C = D) -> (A = B <-> C = D))
2	1	necon3bid 2552	. 2 |- ((A = B <-> C = D) -> (A =/= B <-> C =/= D))
3		id 19	. . 3 |- ((A =/= B <-> C =/= D) -> (A =/= B <-> C =/= D))
4	3	necon4bid 2583	. 2 |- ((A =/= B <-> C =/= D) -> (A = B <-> C = D))
5	2, 4	impbii 180	1 |- ((A = B <-> C = D) <-> (A =/= B <-> C =/= D))

Syntax hints:   <-> wb 176   = 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)