Theorem necon4ad 1629

Description: Contrapositive inference for inequality.

Hypothesis

Ref	Expression
necon4ad.1	|- (ph -> (A =/= B -> -. ps))

Assertion

Ref	Expression
necon4ad	|- (ph -> (ps -> A = B))

Proof of Theorem necon4ad

Step	Hyp	Ref	Expression
1		necon4ad.1	. . 3 |- (ph -> (A =/= B -> -. ps))
2		df-ne 1590	. . 3 |- (A =/= B <-> -. A = B)
3	1, 2	syl5ibr 207	. 2 |- (ph -> (-. A = B -> -. ps))
4	3	a3d 75	1 |- (ph -> (ps -> A = B))

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

This theorem is referenced by:  necon4d 1631  aceq5 4750  sfseqeq 10529

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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