Theorem syl333anc 1265

Description: A syllogism inference combined with contraction. (Contributed by NM, 10-Mar-2012.)

Hypotheses

Ref	Expression
sylXanc.1	|- ( ph -> ps )
sylXanc.2	|- ( ph -> ch )
sylXanc.3	|- ( ph -> th )
sylXanc.4	|- ( ph -> ta )
sylXanc.5	|- ( ph -> et )
sylXanc.6	|- ( ph -> ze )
sylXanc.7	|- ( ph -> si )
sylXanc.8	|- ( ph -> rh )
sylXanc.9	|- ( ph -> mu )
syl333anc.10	|- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et /\ ze ) /\ ( si /\ rh /\ mu ) ) -> la )

Assertion

Ref	Expression
syl333anc	|- ( ph -> la )

Proof of Theorem syl333anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. 2 |- ( ph -> ps )
2		sylXanc.2	. 2 |- ( ph -> ch )
3		sylXanc.3	. 2 |- ( ph -> th )
4		sylXanc.4	. 2 |- ( ph -> ta )
5		sylXanc.5	. 2 |- ( ph -> et )
6		sylXanc.6	. 2 |- ( ph -> ze )
7		sylXanc.7	. . 3 |- ( ph -> si )
8		sylXanc.8	. . 3 |- ( ph -> rh )
9		sylXanc.9	. . 3 |- ( ph -> mu )
10	7, 8, 9	3jca 1172	. 2 |- ( ph -> ( si /\ rh /\ mu ) )
11		syl333anc.10	. 2 |- ( ( ( ps /\ ch /\ th ) /\ ( ta /\ et /\ ze ) /\ ( si /\ rh /\ mu ) ) -> la )
12	1, 2, 3, 4, 5, 6, 10, 11	syl331anc 1258	1 |- ( ph -> la )

Syntax hints:   -> wi 4   /\ w3a 973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 975

This theorem is referenced by: (None)