Theorem necon4bid 2716

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 2713	. 2 |- ( ph -> ( C = D <-> -. A =/= B ) )
3		nne 2658	. 2 |- ( -. A =/= B <-> A = B )
4	2, 3	syl6rbb 262	1 |- ( ph -> ( A = B <-> C = D ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   = wceq 1395   =/= wne 2652

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 185  df-ne 2654

This theorem is referenced by:  nebi  2767  znnenlem  13957  rpexp  14273  norm-i  26173  trlid0b  36046