Theorem pm2.61dane 2093

Description: Deduction eliminating an inequality in an antecedent.

Hypotheses

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

Assertion

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

Proof of Theorem pm2.61dane

Step	Hyp	Ref	Expression
1		pm2.61dane.1	. . 3 |- ((ph /\ A = B) -> ps)
2	1	ex 402	. 2 |- (ph -> (A = B -> ps))
3		pm2.61dane.2	. . 3 |- ((ph /\ A =/= B) -> ps)
4	3	ex 402	. 2 |- (ph -> (A =/= B -> ps))
5	2, 4	pm2.61dne 2091	1 |- (ph -> ps)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   =/= wne 2017

This theorem is referenced by:  mulc1cncf 8541  atcvrj2b 17069  atltcvr 17072  ps2 17079  pmodlem1 17307  pmapjat 17314  osumcl 17375  pexmidOLD 17386

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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