Theorem syl3anl2 872

Description: A syllogism inference.

Hypotheses

Ref	Expression
syl3anl1.1	|- (((ph /\ ps /\ ch) /\ th) -> ta)
syl3anl2.2	|- (et -> ps)

Assertion

Ref	Expression
syl3anl2	|- (((ph /\ et /\ ch) /\ th) -> ta)

Proof of Theorem syl3anl2

Step	Hyp	Ref	Expression
1		syl3anl1.1	. . . 4 |- (((ph /\ ps /\ ch) /\ th) -> ta)
2	1	ex 373	. . 3 |- ((ph /\ ps /\ ch) -> (th -> ta))
3		syl3anl2.2	. . 3 |- (et -> ps)
4	2, 3	syl3an2 859	. 2 |- ((ph /\ et /\ ch) -> (th -> ta))
5	4	imp 350	1 |- (((ph /\ et /\ ch) /\ th) -> ta)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774

This theorem is referenced by:  syl3anr2 876  divsubdirt 5739  ltdiv2t 5843  qbtwnre 6224  dnsconst 7738  ssblex 7808  homco2t 9840

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 776

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