Theorem syl233anc 1355

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 )
syl233anc.8	|- ( ph -> rh )
syl233anc.9	|- ( ( ( ps /\ ch ) /\ ( th /\ ta /\ et ) /\ ( ze /\ si /\ rh ) ) -> mu )

Assertion

Ref	Expression
syl233anc	|- ( ph -> mu )

Proof of Theorem syl233anc

Step	Hyp	Ref	Expression
1		syl12anc.1	. . 3 |- ( ph -> ps )
2		syl12anc.2	. . 3 |- ( ph -> ch )
3	1, 2	jca 554	. 2 |- ( ph -> ( ps /\ ch ) )
4		syl12anc.3	. 2 |- ( ph -> th )
5		syl22anc.4	. 2 |- ( ph -> ta )
6		syl23anc.5	. 2 |- ( ph -> et )
7		syl33anc.6	. 2 |- ( ph -> ze )
8		syl133anc.7	. 2 |- ( ph -> si )
9		syl233anc.8	. 2 |- ( ph -> rh )
10		syl233anc.9	. 2 |- ( ( ( ps /\ ch ) /\ ( th /\ ta /\ et ) /\ ( ze /\ si /\ rh ) ) -> mu )
11	3, 4, 5, 6, 7, 8, 9, 10	syl133anc 1349	1 |- ( ph -> mu )

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:  br8d  29422  2llnjN  34853  cdleme16b  35566  cdleme18d  35582  cdleme19d  35594  cdleme20bN  35598  cdleme20l1  35608  cdleme22cN  35630  cdleme22eALTN  35633  cdleme22f  35634  cdlemg33c0  35990  cdlemk5  36124  cdlemk5u  36149  cdlemky  36214  cdlemkyyN  36250