Theorem syl3an 1224

Description: A triple syllogism inference. (Contributed by NM, 13-May-2004.)

Hypotheses

Ref	Expression
syl3an.1	|- (ph -> ps)
syl3an.2	|- (ch -> th)
syl3an.3	|- (ta -> et)
syl3an.4	|- ((ps /\ th /\ et) -> ze)

Assertion

Ref	Expression
syl3an	|- ((ph /\ ch /\ ta) -> ze)

Proof of Theorem syl3an

Step	Hyp	Ref	Expression
1		syl3an.1	. . 3 |- (ph -> ps)
2		syl3an.2	. . 3 |- (ch -> th)
3		syl3an.3	. . 3 |- (ta -> et)
4	1, 2, 3	3anim123i 1137	. 2 |- ((ph /\ ch /\ ta) -> (ps /\ th /\ et))
5		syl3an.4	. 2 |- ((ps /\ th /\ et) -> ze)
6	4, 5	syl 15	1 |- ((ph /\ ch /\ ta) -> ze)

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:  peano5  4410  spfininduct  4541  eloprabga  5579  clos1induct  5881