Theorem syl213anc

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 )
	syl23anc.5	|- ( ph -> et )
	syl33anc.6	|- ( ph -> ze )
	syl213anc.7	|- ( ( ( ps /\ ch ) /\ th /\ ( ta /\ et /\ ze ) ) -> si )
Assertion	syl213anc	|- ( ph -> si )

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		syl23anc.5	|- ( ph -> et )
6		syl33anc.6	|- ( ph -> ze )
7		syl213anc.7	|- ( ( ( ps /\ ch ) /\ th /\ ( ta /\ et /\ ze ) ) -> si )
8	1 2	jca	|- ( ph -> ( ps /\ ch ) )
9	8 3 4 5 6 7	syl113anc	|- ( ph -> si )