Theorem syl3anb

Description: A triple syllogism inference. (Contributed by NM, 15-Oct-2005)

	Ref	Expression
Hypotheses	syl3anb.1	|- ( ph <-> ps )
	syl3anb.2	|- ( ch <-> th )
	syl3anb.3	|- ( ta <-> et )
	syl3anb.4	|- ( ( ps /\ th /\ et ) -> ze )
Assertion	syl3anb	|- ( ( ph /\ ch /\ ta ) -> ze )

Proof

Step	Hyp	Ref	Expression
1		syl3anb.1	|- ( ph <-> ps )
2		syl3anb.2	|- ( ch <-> th )
3		syl3anb.3	|- ( ta <-> et )
4		syl3anb.4	|- ( ( ps /\ th /\ et ) -> ze )
5	1 2 3	3anbi123i	|- ( ( ph /\ ch /\ ta ) <-> ( ps /\ th /\ et ) )
6	5 4	sylbi	|- ( ( ph /\ ch /\ ta ) -> ze )