Theorem syl121anc 1283

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 )
syl121anc.5	|- ( ( ps /\ ( ch /\ th ) /\ ta ) -> et )

Assertion

Ref	Expression
syl121anc	|- ( ph -> et )

Proof of Theorem syl121anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. 2 |- ( ph -> ps )
2		sylXanc.2	. . 3 |- ( ph -> ch )
3		sylXanc.3	. . 3 |- ( ph -> th )
4	2, 3	jca 306	. 2 |- ( ph -> ( ch /\ th ) )
5		sylXanc.4	. 2 |- ( ph -> ta )
6		syl121anc.5	. 2 |- ( ( ps /\ ( ch /\ th ) /\ ta ) -> et )
7	1, 4, 5, 6	syl3anc 1278	1 |- ( ph -> et )

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:  syl122anc  1287  tfisi  4734  tfrcllemsucfn  6624  sbthlemi6  7279  sbthlemi8  7281  div32apd  9147  div13apd  9148  expdivapd  11140  swrdsbslen  11454  modfsummodlemstep  12243  pcqmul  13105  pcid  13126  pcneg  13127  pc2dvds  13132  pcz  13134  pcaddlem  13141  pcadd  13142  pcmpt2  13146  pcbc  13153  qexpz  13154  expnprm  13155  ennnfonelemg  13346  ssblex  15623  bcmono  16265  depind  16916