Theorem syl113anc 1194

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)
syl113anc.6	|- ((ps /\ ch /\ (th /\ ta /\ et)) -> ze)

Assertion

Ref	Expression
syl113anc	|- (ph -> ze)

Proof of Theorem syl113anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. 2 |- (ph -> ps)
2		sylXanc.2	. 2 |- (ph -> ch)
3		sylXanc.3	. . 3 |- (ph -> th)
4		sylXanc.4	. . 3 |- (ph -> ta)
5		sylXanc.5	. . 3 |- (ph -> et)
6	3, 4, 5	3jca 1132	. 2 |- (ph -> (th /\ ta /\ et))
7		syl113anc.6	. 2 |- ((ps /\ ch /\ (th /\ ta /\ et)) -> ze)
8	1, 2, 6, 7	syl3anc 1182	1 |- (ph -> ze)

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:  syl123anc  1199  syl213anc  1201