Theorem syl12anc 1180

Description: Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)

Hypotheses

Ref	Expression
sylXanc.1	|- (ph -> ps)
sylXanc.2	|- (ph -> ch)
sylXanc.3	|- (ph -> th)
syl12anc.4	|- ((ps /\ (ch /\ th)) -> ta)

Assertion

Ref	Expression
syl12anc	|- (ph -> ta)

Proof of Theorem syl12anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. . 3 |- (ph -> ps)
2		sylXanc.2	. . 3 |- (ph -> ch)
3		sylXanc.3	. . 3 |- (ph -> th)
4	1, 2, 3	jca32 521	. 2 |- (ph -> (ps /\ (ch /\ th)))
5		syl12anc.4	. 2 |- ((ps /\ (ch /\ th)) -> ta)
6	4, 5	syl 15	1 |- (ph -> ta)

Syntax hints:   -> wi 4   /\ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by:  syl22anc  1183  raaan  3658  raaanv  3659  nndisjeq  4430  prepeano4  4452  ssfin  4471  ncfinraise  4482  ncfinlower  4484  nnpweq  4524  peano4  4558  f1oiso2  5501  frecsuc  6323