Theorem necon1ad 1623

Description: Contrapositive deduction for inequality.

Hypothesis

Ref	Expression
necon1ad.1	|- (ph -> (-. ps -> A = B))

Assertion

Ref	Expression
necon1ad	|- (ph -> (A =/= B -> ps))

Proof of Theorem necon1ad

Step	Hyp	Ref	Expression
1		necon1ad.1	. . 3 |- (ph -> (-. ps -> A = B))
2	1	con1d 93	. 2 |- (ph -> (-. A = B -> ps))
3		df-ne 1579	. 2 |- (A =/= B <-> -. A = B)
4	2, 3	syl5ib 206	1 |- (ph -> (A =/= B -> ps))

Syntax hints:  -. wn 2   -> wi 3   = wceq 953   =/= wne 1577

This theorem is referenced by:  onmindif2 3051  aceq5lem4 4710  dfn2 6059  uzwo4OLD 6158  uzwo 6387  h1datom 9421  atsseq 10182

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 1579

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