Theorem pm2.61dane 2595

Description: Deduction eliminating an inequality in an antecedent. (Contributed by NM, 30-Nov-2011.)

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 423	. 2 |- (ph -> (A = B -> ps))
3		pm2.61dane.2	. . 3 |- ((ph /\ A =/= B) -> ps)
4	3	ex 423	. 2 |- (ph -> (A =/= B -> ps))
5	2, 4	pm2.61dne 2594	1 |- (ph -> ps)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   =/= wne 2517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360  df-ne 2519

This theorem is referenced by:  pm2.61da2ne  2596  pm2.61da3ne  2597