Theorem syl122anc 1259

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 1255	1 |- ( ph -> ze )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981

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 983

This theorem is referenced by:  divdiv32apd  8924  divcanap5d  8925  divcanap7d  8927  divdivap1d  8930  divdivap2d  8931  seq3coll  11024  cau3lem  11540  summodclem2a  11807  prodmodclem2a  12002  prmind2  12557  divnumden  12633  pceulem  12732  pcqmul  12741  pcqdiv  12745  pcexp  12747  pcaddlem  12777  pcbc  12789  abladdsub4  13765  ablpnpcan  13771  lmodvs1  14193  blss2ps  14993  blss2  14994  blssps  15014  blss  15015  xmeter  15023  lgsdi  15629