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  6579  tfrcl  6635  mkvprop  7498  ccfunen  7630  caucvgprprlemval  8055  suplocsrlem  8175  peano5uzti  9754  seqf1oglem2  10957  zfz1iso  11293  wrd2ind  11495  lcmneg  12852  prmind2  12898  pcfac  13129  cnmpt12  15388  cnmpt22  15395  limccnp2lem  15777  2sqlem6  16239  2sqlem8  16242  gropd  16288  grstructd2dom  16289  sbthom  17071