Theorem syl211anc 1141

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 )
syl211anc.5	|- ( ( ( ps /\ ch ) /\ th /\ ta ) -> et )

Assertion

Ref	Expression
syl211anc	|- ( ph -> et )

Proof of Theorem syl211anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. . 3 |- ( ph -> ps )
2		sylXanc.2	. . 3 |- ( ph -> ch )
3	1, 2	jca 290	. 2 |- ( ph -> ( ps /\ ch ) )
4		sylXanc.3	. 2 |- ( ph -> th )
5		sylXanc.4	. 2 |- ( ph -> ta )
6		syl211anc.5	. 2 |- ( ( ( ps /\ ch ) /\ th /\ ta ) -> et )
7	3, 4, 5, 6	syl3anc 1135	1 |- ( ph -> et )

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 885

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110  df-3an 887

This theorem is referenced by:  syl212anc  1145  syl221anc  1146