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  8963  divcanap5d  8964  divcanap7d  8966  divdivap1d  8969  divdivap2d  8970  seq3coll  11064  cau3lem  11625  summodclem2a  11892  prodmodclem2a  12087  prmind2  12642  divnumden  12718  pceulem  12817  pcqmul  12826  pcqdiv  12830  pcexp  12832  pcaddlem  12862  pcbc  12874  abladdsub4  13851  ablpnpcan  13857  lmodvs1  14280  blss2ps  15080  blss2  15081  blssps  15101  blss  15102  xmeter  15110  lgsdi  15716