Theorem syl223anc 1352

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 )
syl223anc.8	|- ( ( ( ps /\ ch ) /\ ( th /\ ta ) /\ ( et /\ ze /\ si ) ) -> rh )

Assertion

Ref	Expression
syl223anc	|- ( ph -> rh )

Proof of Theorem syl223anc

Step	Hyp	Ref	Expression
1		syl12anc.1	. 2 |- ( ph -> ps )
2		syl12anc.2	. 2 |- ( ph -> ch )
3		syl12anc.3	. . 3 |- ( ph -> th )
4		syl22anc.4	. . 3 |- ( ph -> ta )
5	3, 4	jca 554	. 2 |- ( ph -> ( th /\ ta ) )
6		syl23anc.5	. 2 |- ( ph -> et )
7		syl33anc.6	. 2 |- ( ph -> ze )
8		syl133anc.7	. 2 |- ( ph -> si )
9		syl223anc.8	. 2 |- ( ( ( ps /\ ch ) /\ ( th /\ ta ) /\ ( et /\ ze /\ si ) ) -> rh )
10	1, 2, 5, 6, 7, 8, 9	syl213anc 1345	1 |- ( ph -> rh )

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:  cdleme17d1  35576  cdlemednpq  35586  cdleme19d  35594  cdleme20aN  35597  cdleme20c  35599  cdleme20f  35602  cdleme20g  35603  cdleme20j  35606  cdleme20l1  35608  cdleme20l2  35609  cdlemky  36214  cdlemkyyN  36250