Theorem syl3an3 621

Description: A syllogism inference.

Hypotheses

Ref	Expression
syl3an.1	|- ((ph /\ ps /\ ch) -> th)
syl3an3.2	|- (ta -> ch)

Assertion

Ref	Expression
syl3an3	|- ((ph /\ ps /\ ta ) -> th)

Proof of Theorem syl3an3

Step	Hyp	Ref	Expression
1		syl3an.1	. . . 4 |- ((ph /\ ps /\ ch) -> th)
2	1	3exp 611	. . 3 |- (ph -> (ps -> (ch -> th)))
3		syl3an3.2	. . 3 |- (ta -> ch)
4	2, 3	syl7 24	. 2 |- (ph -> (ps -> (ta -> th)))
5	4	3imp 608	1 |- ((ph /\ ps /\ ta ) -> th)

Syntax hints:   -> wi 2   /\ w3a 581

This theorem is referenced by:  syl3an3b 624  syl3an3br 627  oprabval4g 3053  ecopoprtrn 3247  addsubasst 4150  lesub1t 4352  uzwo3lem1 4614  hvaddsub12t 5015  hvaddsubasst 5018  mdsymlem5 5780

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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