Theorem syl3anl 1377

Description: A triple syllogism inference. (Contributed by NM, 24-Dec-2006.)

Hypotheses

Ref	Expression
syl3anl.1	|- ( ph -> ps )
syl3anl.2	|- ( ch -> th )
syl3anl.3	|- ( ta -> et )
syl3anl.4	|- ( ( ( ps /\ th /\ et ) /\ ze ) -> si )

Assertion

Ref	Expression
syl3anl	|- ( ( ( ph /\ ch /\ ta ) /\ ze ) -> si )

Proof of Theorem syl3anl

Step	Hyp	Ref	Expression
1		syl3anl.1	. . 3 |- ( ph -> ps )
2		syl3anl.2	. . 3 |- ( ch -> th )
3		syl3anl.3	. . 3 |- ( ta -> et )
4	1, 2, 3	3anim123i 1247	. 2 |- ( ( ph /\ ch /\ ta ) -> ( ps /\ th /\ et ) )
5		syl3anl.4	. 2 |- ( ( ( ps /\ th /\ et ) /\ ze ) -> si )
6	4, 5	sylan 488	1 |- ( ( ( ph /\ ch /\ ta ) /\ ze ) -> si )

Syntax hints:   -> wi 4   /\ wa 384   /\ 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:  chlej1  28369  chlej2  28370  atcvatlem  29244