Theorem syl3an9b 891

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 540	. . 3 |- ((ph /\ th) -> (ps <-> ta))
4		syl3an9b.3	. . 3 |- (et -> (ta <-> ze))
5	3, 4	sylan9bb 540	. 2 |- (((ph /\ th) /\ et) -> (ps <-> ze))
6	5	3impa 828	1 |- ((ph /\ th /\ et) -> (ps <-> ze))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775

This theorem is referenced by:  eloprabg 4007  caoprass 4054  caoprdistr 4059  ertr 4274  elo 10444

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 777

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