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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117  df-3an 1011

This theorem is used by:  divdiv32apd  9149  divcanap5d  9150  divcanap7d  9152  divdivap1d  9155  divdivap2d  9156  seq3coll  11310  cau3lem  11897  summodclem2a  12167  prodmodclem2a  12362  prmind2  12917  divnumden  12995  pceulem  13096  pcqmul  13105  pcqdiv  13109  pcexp  13111  pcaddlem  13141  pcbc  13153  abladdsub4  14202  ablpnpcan  14208  gsummptfidmadd  14245  lmodvs1  14737  blss2ps  15598  blss2  15599  blssps  15619  blss  15620  xmeter  15628  lgsdi  16322