Theorem syl3an9b 634

Description: Nested syllogism inference conjoining 3 dissimilar antecedents.

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 418	. . 3 |- ((ph /\ th) -> (ps <-> ta ))
4		syl3an9b.3	. . 3 |- (et -> (ta <-> ze))
5	3, 4	sylan9bb 418	. 2 |- (((ph /\ th) /\ et) -> (ps <-> ze))
6	5	3impa 609	1 |- ((ph /\ th /\ et) -> (ps <-> ze))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   /\ w3a 581

This theorem is referenced by:  eloprabg 3035  caoprass 3068  caoprdistr 3073  ertr 3211

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583

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