Theorem syl31anc 1185

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)
syl31anc.5	|- (((ps /\ ch /\ th) /\ ta) -> et)

Assertion

Ref	Expression
syl31anc	|- (ph -> et)

Proof of Theorem syl31anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. . 3 |- (ph -> ps)
2		sylXanc.2	. . 3 |- (ph -> ch)
3		sylXanc.3	. . 3 |- (ph -> th)
4	1, 2, 3	3jca 1132	. 2 |- (ph -> (ps /\ ch /\ th))
5		sylXanc.4	. 2 |- (ph -> ta)
6		syl31anc.5	. 2 |- (((ps /\ ch /\ th) /\ ta) -> et)
7	4, 5, 6	syl2anc 642	1 |- (ph -> et)

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:  syl32anc  1190  ltfinirr  4458  vfin1cltv  4548  leconnnc  6219  tlecg  6231