Theorem syld3an1 871

Description: A syllogism inference.

Hypotheses

Ref	Expression
syl3an.1	|- ((ph /\ ps /\ ch) -> th)
syld3an1.2	|- ((ta /\ ps /\ ch) -> ph)

Assertion

Ref	Expression
syld3an1	|- ((ta /\ ps /\ ch) -> th)

Proof of Theorem syld3an1

Step	Hyp	Ref	Expression
1		syl3an.1	. . . 4 |- ((ph /\ ps /\ ch) -> th)
2	1	3com13 838	. . 3 |- ((ch /\ ps /\ ph) -> th)
3		syld3an1.2	. . . 4 |- ((ta /\ ps /\ ch) -> ph)
4	3	3com13 838	. . 3 |- ((ch /\ ps /\ ta) -> ph)
5	2, 4	syld3an3 870	. 2 |- ((ch /\ ps /\ ta) -> th)
6	5	3com13 838	1 |- ((ta /\ ps /\ ch) -> th)

Syntax hints:   -> wi 3   /\ w3a 775

This theorem is referenced by:  npncant 5400  ppncant 5481

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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