Theorem syl122anc 1287

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 1283	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:  divdiv32apd  9136  divcanap5d  9137  divcanap7d  9139  divdivap1d  9142  divdivap2d  9143  seq3coll  11272  cau3lem  11858  summodclem2a  12126  prodmodclem2a  12321  prmind2  12876  divnumden  12952  pceulem  13051  pcqmul  13060  pcqdiv  13064  pcexp  13066  pcaddlem  13096  pcbc  13108  abladdsub4  14095  ablpnpcan  14101  gsummptfidmadd  14138  lmodvs1  14625  blss2ps  15430  blss2  15431  blssps  15451  blss  15452  xmeter  15460  lgsdi  16070