Theorem sylsyld 58

Description: A double syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.)

Hypotheses

Ref	Expression
sylsyld.1	|- ( ph -> ps )
sylsyld.2	|- ( ph -> ( ch -> th ) )
sylsyld.3	|- ( ps -> ( th -> ta ) )

Assertion

Ref	Expression
sylsyld	|- ( ph -> ( ch -> ta ) )

Proof of Theorem sylsyld

Step	Hyp	Ref	Expression
1		sylsyld.2	. 2 |- ( ph -> ( ch -> th ) )
2		sylsyld.1	. . 3 |- ( ph -> ps )
3		sylsyld.3	. . 3 |- ( ps -> ( th -> ta ) )
4	2, 3	syl 14	. 2 |- ( ph -> ( th -> ta ) )
5	1, 4	syld 45	1 |- ( ph -> ( ch -> ta ) )

Syntax hints:   -> wi 4

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

This theorem is referenced by:  ax10o  1693  a16g  1836  trintssm  4042  funimaexglem  5206  smoiun  6198  findcard2  6783  ctssdc  6998  mkvprop  7032  ltexprlemrl  7418  archsr  7590  elfz0ubfz0  9902  ctinf  11943