Theorem syl3anb 1225

Description: A triple syllogism inference. (Contributed by NM, 15-Oct-2005.)

Hypotheses

Ref	Expression
syl3anb.1	|- (ph <-> ps)
syl3anb.2	|- (ch <-> th)
syl3anb.3	|- (ta <-> et)
syl3anb.4	|- ((ps /\ th /\ et) -> ze)

Assertion

Ref	Expression
syl3anb	|- ((ph /\ ch /\ ta) -> ze)

Proof of Theorem syl3anb

Step	Hyp	Ref	Expression
1		syl3anb.1	. . 3 |- (ph <-> ps)
2		syl3anb.2	. . 3 |- (ch <-> th)
3		syl3anb.3	. . 3 |- (ta <-> et)
4	1, 2, 3	3anbi123i 1140	. 2 |- ((ph /\ ch /\ ta) <-> (ps /\ th /\ et))
5		syl3anb.4	. 2 |- ((ps /\ th /\ et) -> ze)
6	4, 5	sylbi 187	1 |- ((ph /\ ch /\ ta) -> ze)

Syntax hints:   -> wi 4   <-> wb 176   /\ 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:  syl3anbr  1226