Theorem syl23anc 1189

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)
syl23anc.6	|- (((ps /\ ch) /\ (th /\ ta /\ et)) -> ze)

Assertion

Ref	Expression
syl23anc	|- (ph -> ze)

Proof of Theorem syl23anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. . 3 |- (ph -> ps)
2		sylXanc.2	. . 3 |- (ph -> ch)
3	1, 2	jca 518	. 2 |- (ph -> (ps /\ ch))
4		sylXanc.3	. 2 |- (ph -> th)
5		sylXanc.4	. 2 |- (ph -> ta)
6		sylXanc.5	. 2 |- (ph -> et)
7		syl23anc.6	. 2 |- (((ps /\ ch) /\ (th /\ ta /\ et)) -> ze)
8	3, 4, 5, 6, 7	syl13anc 1184	1 |- (ph -> ze)

Syntax hints:   -> wi 4   /\ wa 358   /\ 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: (None)