Theorem syl3anl 875

Description: A triple syllogism inference.

Hypotheses

Ref	Expression
syl3anl1.1	|- (((ph /\ ps /\ ch) /\ th) -> ta)
syl3anl.2	|- (et -> ph)
syl3anl.3	|- (ze -> ps)
syl3anl.4	|- (si -> ch)

Assertion

Ref	Expression
syl3anl	|- (((et /\ ze /\ si) /\ th) -> ta)

Proof of Theorem syl3anl

Step	Hyp	Ref	Expression
1		syl3anl1.1	. 2 |- (((ph /\ ps /\ ch) /\ th) -> ta)
2		syl3anl.2	. . 3 |- (et -> ph)
3		syl3anl.3	. . 3 |- (ze -> ps)
4		syl3anl.4	. . 3 |- (si -> ch)
5	2, 3, 4	3anim123i 820	. 2 |- ((et /\ ze /\ si) -> (ph /\ ps /\ ch))
6	1, 5	sylan 448	1 |- (((et /\ ze /\ si) /\ th) -> ta)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774

This theorem is referenced by:  nnmcan 4241  mulc1cncf 7231  efsubt 7330  metcnco 7859  htthlem8 8585  chlej1t 9388  chlej2t 9389  atcvatlem 10268

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 776

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