Theorem syl31anc

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 )
	syl3Xanc.4	|- ( ph -> ta )
	syl31anc.5	|- ( ( ( ps /\ ch /\ th ) /\ ta ) -> et )
Assertion	syl31anc	|- ( ph -> et )

Proof

Step	Hyp	Ref	Expression
1		syl3anc.1	|- ( ph -> ps )
2		syl3anc.2	|- ( ph -> ch )
3		syl3anc.3	|- ( ph -> th )
4		syl3Xanc.4	|- ( ph -> ta )
5		syl31anc.5	|- ( ( ( ps /\ ch /\ th ) /\ ta ) -> et )
6	1 2 3	3jca	|- ( ph -> ( ps /\ ch /\ th ) )
7	6 4 5	syl2anc	|- ( ph -> et )