Theorem syl13anc 1184

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)
syl13anc.5	|- ((ps /\ (ch /\ th /\ ta)) -> et)

Assertion

Ref	Expression
syl13anc	|- (ph -> et)

Proof of Theorem syl13anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. 2 |- (ph -> ps)
2		sylXanc.2	. . 3 |- (ph -> ch)
3		sylXanc.3	. . 3 |- (ph -> th)
4		sylXanc.4	. . 3 |- (ph -> ta)
5	2, 3, 4	3jca 1132	. 2 |- (ph -> (ch /\ th /\ ta))
6		syl13anc.5	. 2 |- ((ps /\ (ch /\ th /\ ta)) -> et)
7	1, 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:  syl23anc  1189  syl33anc  1197  pm2.61da3ne  2597  sbthlem3  6206