Theorem syl312anc 1254

Description: Syllogism combined with contraction. (Contributed by NM, 11-Jul-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 )
syl312anc.7	|- ( ( ( ps /\ ch /\ th ) /\ ta /\ ( et /\ ze ) ) -> si )

Assertion

Ref	Expression
syl312anc	|- ( ph -> si )

Proof of Theorem syl312anc

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	. . 3 |- ( ph -> et )
6		sylXanc.6	. . 3 |- ( ph -> ze )
7	5, 6	jca 304	. 2 |- ( ph -> ( et /\ ze ) )
8		syl312anc.7	. 2 |- ( ( ( ps /\ ch /\ th ) /\ ta /\ ( et /\ ze ) ) -> si )
9	1, 2, 3, 4, 7, 8	syl311anc 1247	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:  pythagtriplem19  12236