Theorem necon4bid 2669

Description: Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.)

Hypothesis

Ref	Expression
necon4bid.1	|- ( ph -> ( A =/= B <-> C =/= D ) )

Assertion

Ref	Expression
necon4bid	|- ( ph -> ( A = B <-> C = D ) )

Proof of Theorem necon4bid

Step	Hyp	Ref	Expression
1		necon4bid.1	. . 3 |- ( ph -> ( A =/= B <-> C =/= D ) )
2	1	necon2bbid 2667	. 2 |- ( ph -> ( C = D <-> -. A =/= B ) )
3		nne 2628	. 2 |- ( -. A =/= B <-> A = B )
4	2, 3	syl6rbb 266	1 |- ( ph -> ( A = B <-> C = D ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   = wceq 1444   =/= wne 2622

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 189  df-ne 2624

This theorem is referenced by:  nebi  2704  znnenlem  14264  rpexp  14672  norm-i  26782  trlid0b  33744