Theorem necon4d 1631

Description: Contrapositive inference for inequality.

Hypothesis

Ref	Expression
necon4d.1	|- (ph -> (A =/= B -> C =/= D))

Assertion

Ref	Expression
necon4d	|- (ph -> (C = D -> A = B))

Proof of Theorem necon4d

Step	Hyp	Ref	Expression
1		necon4d.1	. . 3 |- (ph -> (A =/= B -> C =/= D))
2		df-ne 1590	. . 3 |- (C =/= D <-> -. C = D)
3	1, 2	syl6ib 212	. 2 |- (ph -> (A =/= B -> -. C = D))
4	3	necon4ad 1629	1 |- (ph -> (C = D -> A = B))

Syntax hints:  -. wn 2   -> wi 3   = wceq 958   =/= wne 1588

This theorem is referenced by:  oa00 4199  zfregs 4657  expcant 6602  nn0opth 6667  sineq0 8708  his6t 8960

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-ne 1590

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