Theorem syl3c 63

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 62	. 2 |- ( ph -> ( th -> ta ) )
6	1, 5	mpd 13	1 |- ( ph -> 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:  bilukdc  1415  disjiun  4038  tfrlem1  6393  tfrcl  6449  mkvprop  7259  ccfunen  7375  caucvgprprlemval  7800  suplocsrlem  7920  peano5uzti  9480  seqf1oglem2  10663  zfz1iso  10984  lcmneg  12367  prmind2  12413  pcfac  12644  cnmpt12  14730  cnmpt22  14737  limccnp2lem  15119  2sqlem6  15568  2sqlem8  15571  gropd  15615  grstructd2dom  15616  sbthom  15927