Theorem necon4bid 1630

Description: Contrapositive law deduction for inequality.

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 1623	. 2 |- (ph -> (C = D <-> -. A =/= B))
3		nne 1589	. 2 |- (-. A =/= B <-> A = B)
4	2, 3	syl6rbb 537	1 |- (ph -> (A = B <-> C = D))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   = wceq 956   =/= wne 1585

This theorem is referenced by:  msq11 5883  znnenlem 7501  norm-it 8996

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 147  df-an 225  df-ne 1587

Copyright terms: Public domain