Theorem pm2.61dne 1638

Description: Deduction eliminating an inequality in an antecedent.

Hypotheses

Ref	Expression
pm2.61dne.1	|- (ph -> (A = B -> ps))
pm2.61dne.2	|- (ph -> (A =/= B -> ps))

Assertion

Ref	Expression
pm2.61dne	|- (ph -> ps)

Proof of Theorem pm2.61dne

Step	Hyp	Ref	Expression
1		pm2.61dne.1	. 2 |- (ph -> (A = B -> ps))
2		pm2.61dne.2	. . 3 |- (ph -> (A =/= B -> ps))
3		df-ne 1590	. . 3 |- (A =/= B <-> -. A = B)
4	2, 3	syl5ibr 207	. 2 |- (ph -> (-. A = B -> ps))
5	1, 4	pm2.61d 127	1 |- (ph -> ps)

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

This theorem is referenced by:  wefrc 2949  oe0lem 4158  efifolem7 8723

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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