Theorem syl121anc 1255

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 1250	1 |- ( ph -> et )

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:  syl122anc  1259  tfisi  4653  tfrcllemsucfn  6462  sbthlemi6  7090  sbthlemi8  7092  div32apd  8922  div13apd  8923  expdivapd  10869  swrdsbslen  11157  modfsummodlemstep  11883  pcqmul  12741  pcid  12762  pcneg  12763  pc2dvds  12768  pcz  12770  pcaddlem  12777  pcadd  12778  pcmpt2  12782  pcbc  12789  qexpz  12790  expnprm  12791  ennnfonelemg  12889  ssblex  15018