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 )

This proof depends on syntax axioms:   -> wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  bilukdc  1445  disjiun  4125  tfrlem1  6583  tfrcl  6639  mkvprop  7504  ccfunen  7636  caucvgprprlemval  8061  suplocsrlem  8181  peano5uzti  9765  seqf1oglem2  10978  zfz1iso  11315  wrd2ind  11517  lcmneg  12877  prmind2  12923  pcfac  13158  cnmpt12  15513  cnmpt22  15520  limccnp2lem  15902  2sqlem6  16444  2sqlem8  16447  gropd  16493  grstructd2dom  16494  sbthom  17277