Theorem syl3an9b

Description: Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995)

	Ref	Expression
Hypotheses	syl3an9b.1	|- ( ph -> ( ps <-> ch ) )
	syl3an9b.2	|- ( th -> ( ch <-> ta ) )
	syl3an9b.3	|- ( et -> ( ta <-> ze ) )
Assertion	syl3an9b	|- ( ( ph /\ th /\ et ) -> ( ps <-> ze ) )

Proof

Step	Hyp	Ref	Expression
1		syl3an9b.1	|- ( ph -> ( ps <-> ch ) )
2		syl3an9b.2	|- ( th -> ( ch <-> ta ) )
3		syl3an9b.3	|- ( et -> ( ta <-> ze ) )
4	1 2	sylan9bb	|- ( ( ph /\ th ) -> ( ps <-> ta ) )
5	4 3	sylan9bb	|- ( ( ( ph /\ th ) /\ et ) -> ( ps <-> ze ) )
6	5	3impa	|- ( ( ph /\ th /\ et ) -> ( ps <-> ze ) )