Theorem syl311anc 1247

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 )
syl311anc.6	|- ( ( ( ps /\ ch /\ th ) /\ ta /\ et ) -> ze )

Assertion

Ref	Expression
syl311anc	|- ( ph -> ze )

Proof of Theorem syl311anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. . 3 |- ( ph -> ps )
2		sylXanc.2	. . 3 |- ( ph -> ch )
3		sylXanc.3	. . 3 |- ( ph -> th )
4	1, 2, 3	3jca 1172	. 2 |- ( ph -> ( ps /\ ch /\ th ) )
5		sylXanc.4	. 2 |- ( ph -> ta )
6		sylXanc.5	. 2 |- ( ph -> et )
7		syl311anc.6	. 2 |- ( ( ( ps /\ ch /\ th ) /\ ta /\ et ) -> ze )
8	4, 5, 6, 7	syl3anc 1233	1 |- ( ph -> ze )

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:  syl312anc  1254  syl321anc  1255  syl313anc  1257  syl331anc  1258  pythagtrip  12237