Theorem syl3anbr 1370

Description: A triple syllogism inference. (Contributed by NM, 29-Dec-2011.)

Hypotheses

Ref	Expression
syl3anbr.1	|- ( ps <-> ph )
syl3anbr.2	|- ( th <-> ch )
syl3anbr.3	|- ( et <-> ta )
syl3anbr.4	|- ( ( ps /\ th /\ et ) -> ze )

Assertion

Ref	Expression
syl3anbr	|- ( ( ph /\ ch /\ ta ) -> ze )

Proof of Theorem syl3anbr

Step	Hyp	Ref	Expression
1		syl3anbr.1	. . 3 |- ( ps <-> ph )
2	1	bicomi 214	. 2 |- ( ph <-> ps )
3		syl3anbr.2	. . 3 |- ( th <-> ch )
4	3	bicomi 214	. 2 |- ( ch <-> th )
5		syl3anbr.3	. . 3 |- ( et <-> ta )
6	5	bicomi 214	. 2 |- ( ta <-> et )
7		syl3anbr.4	. 2 |- ( ( ps /\ th /\ et ) -> ze )
8	2, 4, 6, 7	syl3anb 1369	1 |- ( ( ph /\ ch /\ ta ) -> ze )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037

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 197  df-an 386  df-3an 1039

This theorem is referenced by:  abvtriv  18841  colinearxfr  32182  paddval  35084