Theorem pm2.61neOLD 2088

Description: Deduction eliminating an inequality in an antecedent.

Hypotheses

Ref	Expression
pm2.61ne.1	|- (A = B -> (ps <-> ch))
pm2.61ne.2	|- ((ph /\ A =/= B) -> ps)
pm2.61ne.3	|- (ph -> ch)

Assertion

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

Proof of Theorem pm2.61neOLD

Step	Hyp	Ref	Expression
1		pm2.61ne.1	. . . 4 |- (A = B -> (ps <-> ch))
2		pm2.61ne.3	. . . 4 |- (ph -> ch)
3	1, 2	syl5bir 227	. . 3 |- (A = B -> (ph -> ps))
4	3	impcom 378	. 2 |- ((ph /\ A = B) -> ps)
5		pm2.61ne.2	. . 3 |- ((ph /\ A =/= B) -> ps)
6		df-ne 2019	. . 3 |- (A =/= B <-> -. A = B)
7	5, 6	sylan2br 502	. 2 |- ((ph /\ -. A = B) -> ps)
8	4, 7	pm2.61dan 535	1 |- (ph -> ps)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   =/= wne 2017

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 164  df-an 242  df-ne 2019

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