Theorem syl322anc

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 )
	syl133anc.7	|- ( ph -> si )
	syl322anc.8	|- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et ) /\ ( ze /\ si ) ) -> rh )
Assertion	syl322anc	|- ( ph -> rh )

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		syl133anc.7	|- ( ph -> si )
8		syl322anc.8	|- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et ) /\ ( ze /\ si ) ) -> rh )
9	6 7	jca	|- ( ph -> ( ze /\ si ) )
10	1 2 3 4 5 9 8	syl321anc	|- ( ph -> rh )