Theorem syl3anb 869

Description: A triple syllogism inference.

Hypotheses

Ref	Expression
syl3an.1	|- ((ph /\ ps /\ ch) -> th)
syl3anb.2	|- (ta <-> ph)
syl3anb.3	|- (et <-> ps)
syl3anb.4	|- (ze <-> ch)

Assertion

Ref	Expression
syl3anb	|- ((ta /\ et /\ ze) -> th)

Proof of Theorem syl3anb

Step	Hyp	Ref	Expression
1		syl3anb.2	. . 3 |- (ta <-> ph)
2		syl3anb.3	. . 3 |- (et <-> ps)
3		syl3anb.4	. . 3 |- (ze <-> ch)
4	1, 2, 3	3anbi123i 822	. 2 |- ((ta /\ et /\ ze) <-> (ph /\ ps /\ ch))
5		syl3an.1	. 2 |- ((ph /\ ps /\ ch) -> th)
6	4, 5	sylbi 199	1 |- ((ta /\ et /\ ze) -> th)

Syntax hints:   -> wi 3   <-> wb 146   /\ w3a 775

This theorem is referenced by:  grpsn 8124  ringsn 8163

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 777

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