Theorem syl3anbr 1226

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 193	. 2 |- (ph <-> ps)
3		syl3anbr.2	. . 3 |- (th <-> ch)
4	3	bicomi 193	. 2 |- (ch <-> th)
5		syl3anbr.3	. . 3 |- (et <-> ta)
6	5	bicomi 193	. 2 |- (ta <-> et)
7		syl3anbr.4	. 2 |- ((ps /\ th /\ et) -> ze)
8	2, 4, 6, 7	syl3anb 1225	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: (None)