Theorem syl331anc 1207

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)
syl331anc.8	|- (((ps /\ ch /\ th) /\ (ta /\ et /\ ze) /\ si) -> rh)

Assertion

Ref	Expression
syl331anc	|- (ph -> rh)

Proof of Theorem syl331anc

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	. . 3 |- (ph -> ta)
5		sylXanc.5	. . 3 |- (ph -> et)
6		sylXanc.6	. . 3 |- (ph -> ze)
7	4, 5, 6	3jca 1132	. 2 |- (ph -> (ta /\ et /\ ze))
8		sylXanc.7	. 2 |- (ph -> si)
9		syl331anc.8	. 2 |- (((ps /\ ch /\ th) /\ (ta /\ et /\ ze) /\ si) -> rh)
10	1, 2, 3, 7, 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:  syl332anc  1213  syl333anc  1214