Theorem syl3c 62

Description: A syllogism inference combined with contraction. (Contributed by Alan Sare, 7-Jul-2011.)

Hypotheses

Ref	Expression
syl3c.1	|- ( ph -> ps )
syl3c.2	|- ( ph -> ch )
syl3c.3	|- ( ph -> th )
syl3c.4	|- ( ps -> ( ch -> ( th -> ta ) ) )

Assertion

Ref	Expression
syl3c	|- ( ph -> ta )

Proof of Theorem syl3c

Step	Hyp	Ref	Expression
1		syl3c.3	. 2 |- ( ph -> th )
2		syl3c.1	. . 3 |- ( ph -> ps )
3		syl3c.2	. . 3 |- ( ph -> ch )
4		syl3c.4	. . 3 |- ( ps -> ( ch -> ( th -> ta ) ) )
5	2, 3, 4	sylc 61	. 2 |- ( ph -> ( th -> ta ) )
6	1, 5	mpd 13	1 |- ( ph -> ta )

Syntax hints:   -> wi 4

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

This theorem is referenced by:  bilukdc  1332  disjiun  3840  tfrlem1  6073  tfrcl  6129  caucvgprprlemval  7245  peano5uzti  8852  zfz1iso  10242  lcmneg  11330  prmind2  11376