Theorem syl3anbr

Description: A triple syllogism inference. (Contributed by NM, 29-Dec-2011)

	Ref	Expression
Hypotheses	syl3anbr.1	|- ( ps <-> ph )
	syl3anbr.2	|- ( th <-> ch )
	syl3anbr.3	|- ( et <-> ta )
	syl3anbr.4	|- ( ( ps /\ th /\ et ) -> ze )
Assertion	syl3anbr	|- ( ( ph /\ ch /\ ta ) -> ze )

Proof

Step	Hyp	Ref	Expression
1		syl3anbr.1	|- ( ps <-> ph )
2		syl3anbr.2	|- ( th <-> ch )
3		syl3anbr.3	|- ( et <-> ta )
4		syl3anbr.4	|- ( ( ps /\ th /\ et ) -> ze )
5	1	bicomi	|- ( ph <-> ps )
6	2	bicomi	|- ( ch <-> th )
7	3	bicomi	|- ( ta <-> et )
8	5 6 7 4	syl3anb	|- ( ( ph /\ ch /\ ta ) -> ze )