Theorem pm2.61ne

Description: Deduction eliminating an inequality in an antecedent. (Contributed by NM, 24-May-2006) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 25-Nov-2019)

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

Proof

Step	Hyp	Ref	Expression
1		pm2.61ne.1	|- ( A = B -> ( ps <-> ch ) )
2		pm2.61ne.2	|- ( ( ph /\ A =/= B ) -> ps )
3		pm2.61ne.3	|- ( ph -> ch )
4	3 1	syl5ibr	|- ( A = B -> ( ph -> ps ) )
5	2	expcom	|- ( A =/= B -> ( ph -> ps ) )
6	4 5	pm2.61ine	|- ( ph -> ps )