Theorem syl3anb 1321

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 1219	. 2 |- ( ( ph /\ ch /\ ta ) <-> ( ps /\ th /\ et ) )
5		syl3anb.4	. 2 |- ( ( ps /\ th /\ et ) -> ze )
6	4, 5	sylbi 121	1 |- ( ( ph /\ ch /\ ta ) -> ze )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   /\ w3a 1009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117  df-3an 1011

This theorem is used by:  syl3anbr  1322  poxp  6468  lmodvscl  14641