Theorem syl211anc 1234

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 304	. 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 1228	1 |- ( ph -> et )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968

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 970

This theorem is referenced by:  syl212anc  1238  syl221anc  1239  relogbexpap  13516  rplogbcxp  13521