Theorem syl123anc 1250

Description: Syllogism combined with contraction. (Contributed by NM, 11-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 )
syl123anc.7	|- ( ( ps /\ ( ch /\ th ) /\ ( ta /\ et /\ ze ) ) -> si )

Assertion

Ref	Expression
syl123anc	|- ( ph -> si )

Proof of Theorem syl123anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. 2 |- ( ph -> ps )
2		sylXanc.2	. . 3 |- ( ph -> ch )
3		sylXanc.3	. . 3 |- ( ph -> th )
4	2, 3	jca 304	. 2 |- ( ph -> ( ch /\ th ) )
5		sylXanc.4	. 2 |- ( ph -> ta )
6		sylXanc.5	. 2 |- ( ph -> et )
7		sylXanc.6	. 2 |- ( ph -> ze )
8		syl123anc.7	. 2 |- ( ( ps /\ ( ch /\ th ) /\ ( ta /\ et /\ ze ) ) -> si )
9	1, 4, 5, 6, 7, 8	syl113anc 1245	1 |- ( ph -> si )

Syntax hints:   -> wi 4   /\ wa 103   /\ 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)