Theorem syl3an3 1217

Description: A syllogism inference. (Contributed by NM, 22-Aug-1995.)

Hypotheses

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

Assertion

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

Proof of Theorem syl3an3

Step	Hyp	Ref	Expression
1		syl3an3.1	. . 3 |- (ph -> th)
2		syl3an3.2	. . . 4 |- ((ps /\ ch /\ th) -> ta)
3	2	3exp 1150	. . 3 |- (ps -> (ch -> (th -> ta)))
4	1, 3	syl7 63	. 2 |- (ps -> (ch -> (ph -> ta)))
5	4	3imp 1145	1 |- ((ps /\ ch /\ ph) -> ta)

Syntax hints:   -> wi 4   /\ w3a 934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936

This theorem is referenced by:  syl3an3b  1220  syl3an3br  1223  vtoclgft  2906  fvopab4t  5386