Theorem syl222anc 1198

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)
syl222anc.7	|- (((ps /\ ch) /\ (th /\ ta) /\ (et /\ ze)) -> si)

Assertion

Ref	Expression
syl222anc	|- (ph -> si)

Proof of Theorem syl222anc

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	5, 6	jca 518	. 2 |- (ph -> (et /\ ze))
8		syl222anc.7	. 2 |- (((ps /\ ch) /\ (th /\ ta) /\ (et /\ ze)) -> si)
9	1, 2, 3, 4, 7, 8	syl221anc 1193	1 |- (ph -> si)

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:  3anandis  1283  3anandirs  1284