Theorem syl22anc

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

	Ref	Expression
Hypotheses	syl12anc.1	|- ( ph -> ps )
	syl12anc.2	|- ( ph -> ch )
	syl12anc.3	|- ( ph -> th )
	syl22anc.4	|- ( ph -> ta )
	syl22anc.5	|- ( ( ( ps /\ ch ) /\ ( th /\ ta ) ) -> et )
Assertion	syl22anc	|- ( ph -> et )

Proof

Step	Hyp	Ref	Expression
1		syl12anc.1	|- ( ph -> ps )
2		syl12anc.2	|- ( ph -> ch )
3		syl12anc.3	|- ( ph -> th )
4		syl22anc.4	|- ( ph -> ta )
5		syl22anc.5	|- ( ( ( ps /\ ch ) /\ ( th /\ ta ) ) -> et )
6	1 2	jca	|- ( ph -> ( ps /\ ch ) )
7	6 3 4 5	syl12anc	|- ( ph -> et )