Theorem syl3anc

Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012)

	Ref	Expression
Hypotheses	syl3anc.1	|- ( ph -> ps )
	syl3anc.2	|- ( ph -> ch )
	syl3anc.3	|- ( ph -> th )
	syl3anc.4	|- ( ( ps /\ ch /\ th ) -> ta )
Assertion	syl3anc	|- ( ph -> ta )

Proof

Step	Hyp	Ref	Expression
1		syl3anc.1	|- ( ph -> ps )
2		syl3anc.2	|- ( ph -> ch )
3		syl3anc.3	|- ( ph -> th )
4		syl3anc.4	|- ( ( ps /\ ch /\ th ) -> ta )
5	1 2 3	3jca	|- ( ph -> ( ps /\ ch /\ th ) )
6	5 4	syl	|- ( ph -> ta )