Theorem syl3an1 1215

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

Hypotheses

Ref	Expression
syl3an1.1	|- (ph -> ps)
syl3an1.2	|- ((ps /\ ch /\ th) -> ta)

Assertion

Ref	Expression
syl3an1	|- ((ph /\ ch /\ th) -> ta)

Proof of Theorem syl3an1

Step	Hyp	Ref	Expression
1		syl3an1.1	. . 3 |- (ph -> ps)
2	1	3anim1i 1138	. 2 |- ((ph /\ ch /\ th) -> (ps /\ ch /\ th))
3		syl3an1.2	. 2 |- ((ps /\ ch /\ th) -> ta)
4	2, 3	syl 15	1 |- ((ph /\ ch /\ th) -> 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:  syl3an1b  1218  syl3an1br  1221  ltfintri  4467  lecadd2  6267