Theorem syl131anc 1182

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 )
syl131anc.6	|- ( ( ps /\ ( ch /\ th /\ ta ) /\ et ) -> ze )

Assertion

Ref	Expression
syl131anc	|- ( ph -> ze )

Proof of Theorem syl131anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. 2 |- ( ph -> ps )
2		sylXanc.2	. . 3 |- ( ph -> ch )
3		sylXanc.3	. . 3 |- ( ph -> th )
4		sylXanc.4	. . 3 |- ( ph -> ta )
5	2, 3, 4	3jca 1118	. 2 |- ( ph -> ( ch /\ th /\ ta ) )
6		sylXanc.5	. 2 |- ( ph -> et )
7		syl131anc.6	. 2 |- ( ( ps /\ ( ch /\ th /\ ta ) /\ et ) -> ze )
8	1, 5, 6, 7	syl3anc 1169	1 |- ( ph -> ze )

Syntax hints:   -> wi 4   /\ w3a 919

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115  df-3an 921

This theorem is referenced by:  syl132anc  1187  syl231anc  1189  syl133anc  1192