Theorem syl221anc 1289

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 )
syl221anc.6	|- ( ( ( ps /\ ch ) /\ ( th /\ ta ) /\ et ) -> ze )

Assertion

Ref	Expression
syl221anc	|- ( ph -> ze )

Proof of Theorem syl221anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. 2 |- ( ph -> ps )
2		sylXanc.2	. 2 |- ( ph -> ch )
3		sylXanc.3	. . 3 |- ( ph -> th )
4		sylXanc.4	. . 3 |- ( ph -> ta )
5	3, 4	jca 306	. 2 |- ( ph -> ( th /\ ta ) )
6		sylXanc.5	. 2 |- ( ph -> et )
7		syl221anc.6	. 2 |- ( ( ( ps /\ ch ) /\ ( th /\ ta ) /\ et ) -> ze )
8	1, 2, 5, 6, 7	syl211anc 1284	1 |- ( ph -> ze )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 1011

This theorem is referenced by:  syl222anc  1294  vtocldf  2874  dmdcanapd  9140  exprecap  10995  fzowrddc  11397  xrbdtri  12020  2strbasg  13451  2stropg  13452  fnpr2o  13637  cnptoprest  15263  blssps  15451  blss  15452  metequiv2  15520  xmettx  15534  edgstruct  16219  usgr2v1e2w  16401