Theorem syl3an9b 1250

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 680	. . 3 |- ((ph /\ th) -> (ps <-> ta))
4		syl3an9b.3	. . 3 |- (et -> (ta <-> ze))
5	3, 4	sylan9bb 680	. 2 |- (((ph /\ th) /\ et) -> (ps <-> ze))
6	5	3impa 1146	1 |- ((ph /\ th /\ et) -> (ps <-> ze))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936

This theorem is referenced by:  eloprabg  5580