Theorem syl313anc 1350

Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-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 )
syl133anc.7	|- ( ph -> si )
syl313anc.8	|- ( ( ( ps /\ ch /\ th ) /\ ta /\ ( et /\ ze /\ si ) ) -> rh )

Assertion

Ref	Expression
syl313anc	|- ( ph -> rh )

Proof of Theorem syl313anc

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		syl133anc.7	. . 3 |- ( ph -> si )
8	5, 6, 7	3jca 1242	. 2 |- ( ph -> ( et /\ ze /\ si ) )
9		syl313anc.8	. 2 |- ( ( ( ps /\ ch /\ th ) /\ ta /\ ( et /\ ze /\ si ) ) -> rh )
10	1, 2, 3, 4, 8, 9	syl311anc 1340	1 |- ( ph -> rh )

Syntax hints:   -> wi 4   /\ 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:  syl323anc  1356  osumcllem6N  35247  cdlemg13  35940  cdlemk7u  36158  cdlemk31  36184  cdlemk27-3  36195  cdlemk19ylem  36218  cdlemk46  36236