Theorem syl213anc 1345

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 )
syl213anc.7	|- ( ( ( ps /\ ch ) /\ th /\ ( ta /\ et /\ ze ) ) -> si )

Assertion

Ref	Expression
syl213anc	|- ( ph -> si )

Proof of Theorem syl213anc

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		syl213anc.7	. 2 |- ( ( ( ps /\ ch ) /\ th /\ ( ta /\ et /\ ze ) ) -> si )
9	3, 4, 5, 6, 7, 8	syl113anc 1338	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:  syl223anc  1352  decpmatmul  20577  nosupbnd1lem5  31858