Theorem nebi 2084

Description: Contraposition law for inequality.

Assertion

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

Proof of Theorem nebi

Step	Hyp	Ref	Expression
1		id 73	. . 3 |- ((A = B <-> C = D) -> (A = B <-> C = D))
2	1	necon3bid 2035	. 2 |- ((A = B <-> C = D) -> (A =/= B <-> C =/= D))
3		id 73	. . 3 |- ((A =/= B <-> C =/= D) -> (A =/= B <-> C =/= D))
4	3	necon4bid 2078	. 2 |- ((A =/= B <-> C =/= D) -> (A = B <-> C = D))
5	2, 4	impbii 174	1 |- ((A = B <-> C = D) <-> (A =/= B <-> C =/= D))

Syntax hints:   <-> wb 163   = wceq 1298   =/= wne 2017

This theorem is referenced by:  mulgcdlem2 13757

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 164  df-an 242  df-ne 2019

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