Theorem syl313anc 1206

Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)

Hypotheses

Ref	Expression
sylXanc.1	|- (ph -> ps)
sylXanc.2	|- (ph -> ch)
sylXanc.3	|- (ph -> th)
sylXanc.4	|- (ph -> ta)
sylXanc.5	|- (ph -> et)
sylXanc.6	|- (ph -> ze)
sylXanc.7	|- (ph -> si)
syl313anc.8	|- (((ps /\ ch /\ th) /\ ta /\ (et /\ ze /\ si)) -> rh)

Assertion

Ref	Expression
syl313anc	|- (ph -> rh)

Proof of Theorem syl313anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. 2 |- (ph -> ps)
2		sylXanc.2	. 2 |- (ph -> ch)
3		sylXanc.3	. 2 |- (ph -> th)
4		sylXanc.4	. 2 |- (ph -> ta)
5		sylXanc.5	. . 3 |- (ph -> et)
6		sylXanc.6	. . 3 |- (ph -> ze)
7		sylXanc.7	. . 3 |- (ph -> si)
8	5, 6, 7	3jca 1132	. 2 |- (ph -> (et /\ ze /\ si))
9		syl313anc.8	. 2 |- (((ps /\ ch /\ th) /\ ta /\ (et /\ ze /\ si)) -> rh)
10	1, 2, 3, 4, 8, 9	syl311anc 1196	1 |- (ph -> rh)

Syntax hints:   -> wi 4   /\ w3a 934

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  df-3an 936

This theorem is referenced by:  syl323anc  1212