Theorem syl212anc 1236

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 )
syl212anc.6	|- ( ( ( ps /\ ch ) /\ th /\ ( ta /\ et ) ) -> ze )

Assertion

Ref	Expression
syl212anc	|- ( ph -> ze )

Proof of Theorem syl212anc

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	. . 3 |- ( ph -> ta )
5		sylXanc.5	. . 3 |- ( ph -> et )
6	4, 5	jca 530	. 2 |- ( ph -> ( ta /\ et ) )
7		syl212anc.6	. 2 |- ( ( ( ps /\ ch ) /\ th /\ ( ta /\ et ) ) -> ze )
8	1, 2, 3, 6, 7	syl211anc 1232	1 |- ( ph -> ze )

Syntax hints:   -> wi 4   /\ wa 367   /\ w3a 971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 973

This theorem is referenced by:  rmob  3416  pntrmax  23950  tglineineq  24227  tglineinteq  24229  paddasslem4  35963  4atexlemu  36204  4atexlemv  36205  cdleme20aN  36451  cdleme20g  36457  cdlemg9a  36774  cdlemg12a  36785  cdlemg17dALTN  36806  cdlemg18b  36821  cdlemg18c  36822