Theorem syl3an9b 1300

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

Hypotheses

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

Assertion

Ref	Expression
syl3an9b	|- ( ( ph /\ th /\ et ) -> ( ps <-> ze ) )

Proof of Theorem syl3an9b

Step	Hyp	Ref	Expression
1		syl3an9b.1	. . . 4 |- ( ph -> ( ps <-> ch ) )
2		syl3an9b.2	. . . 4 |- ( th -> ( ch <-> ta ) )
3	1, 2	sylan9bb 458	. . 3 |- ( ( ph /\ th ) -> ( ps <-> ta ) )
4		syl3an9b.3	. . 3 |- ( et -> ( ta <-> ze ) )
5	3, 4	sylan9bb 458	. 2 |- ( ( ( ph /\ th ) /\ et ) -> ( ps <-> ze ) )
6	5	3impa 1184	1 |- ( ( ph /\ th /\ et ) -> ( ps <-> ze ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 970

This theorem is referenced by:  eloprabg  5930