Theorem syl122anc 1280

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 )
syl122anc.6	|- ( ( ps /\ ( ch /\ th ) /\ ( ta /\ et ) ) -> ze )

Assertion

Ref	Expression
syl122anc	|- ( ph -> ze )

Proof of Theorem syl122anc

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 306	. 2 |- ( ph -> ( ta /\ et ) )
7		syl122anc.6	. 2 |- ( ( ps /\ ( ch /\ th ) /\ ( ta /\ et ) ) -> ze )
8	1, 2, 3, 6, 7	syl121anc 1276	1 |- ( ph -> ze )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002

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 1004

This theorem is referenced by:  divdiv32apd  8986  divcanap5d  8987  divcanap7d  8989  divdivap1d  8992  divdivap2d  8993  seq3coll  11096  cau3lem  11665  summodclem2a  11932  prodmodclem2a  12127  prmind2  12682  divnumden  12758  pceulem  12857  pcqmul  12866  pcqdiv  12870  pcexp  12872  pcaddlem  12902  pcbc  12914  abladdsub4  13891  ablpnpcan  13897  lmodvs1  14320  blss2ps  15120  blss2  15121  blssps  15141  blss  15142  xmeter  15150  lgsdi  15756