Theorem syl122anc 1258

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 1254	1 |- ( ph -> ze )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980

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 982

This theorem is referenced by:  divdiv32apd  8888  divcanap5d  8889  divcanap7d  8891  divdivap1d  8894  divdivap2d  8895  seq3coll  10985  cau3lem  11396  summodclem2a  11663  prodmodclem2a  11858  prmind2  12413  divnumden  12489  pceulem  12588  pcqmul  12597  pcqdiv  12601  pcexp  12603  pcaddlem  12633  pcbc  12645  abladdsub4  13621  ablpnpcan  13627  lmodvs1  14049  blss2ps  14849  blss2  14850  blssps  14870  blss  14871  xmeter  14879  lgsdi  15485