Theorem syl312anc 1347

Description: Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)

Hypotheses

Ref	Expression
syl12anc.1	|- ( ph -> ps )
syl12anc.2	|- ( ph -> ch )
syl12anc.3	|- ( ph -> th )
syl22anc.4	|- ( ph -> ta )
syl23anc.5	|- ( ph -> et )
syl33anc.6	|- ( ph -> ze )
syl312anc.7	|- ( ( ( ps /\ ch /\ th ) /\ ta /\ ( et /\ ze ) ) -> si )

Assertion

Ref	Expression
syl312anc	|- ( ph -> si )

Proof of Theorem syl312anc

Step	Hyp	Ref	Expression
1		syl12anc.1	. 2 |- ( ph -> ps )
2		syl12anc.2	. 2 |- ( ph -> ch )
3		syl12anc.3	. 2 |- ( ph -> th )
4		syl22anc.4	. 2 |- ( ph -> ta )
5		syl23anc.5	. . 3 |- ( ph -> et )
6		syl33anc.6	. . 3 |- ( ph -> ze )
7	5, 6	jca 554	. 2 |- ( ph -> ( et /\ ze ) )
8		syl312anc.7	. 2 |- ( ( ( ps /\ ch /\ th ) /\ ta /\ ( et /\ ze ) ) -> si )
9	1, 2, 3, 4, 7, 8	syl311anc 1340	1 |- ( ph -> si )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037

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 197  df-an 386  df-3an 1039

This theorem is referenced by:  pythagtriplem19  15538  cdleme27cl  35654  cdlemefs27cl  35701  cdleme32fvcl  35728  cdlemg16ALTN  35946  cdlemg27a  35980  cdlemg31c  35987  cdlemg39  36004  cdlemk11ta  36217  cdlemk19ylem  36218  cdlemk11tc  36233  cdlemk45  36235  dihmeetlem12N  36607  dihjatc  36706