Theorem syl3c 57

Description: A syllogism inference combined with contraction. e111 without virtual deductions. (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 56	. 2 |- (ph -> (th -> ta))
6	1, 5	mpd 14	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:  extd  5924  symd  5925  nchoicelem19  6308