Theorem syl121anc 1243

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 1238	1 |- ( ph -> et )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978

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 980

This theorem is referenced by:  syl122anc  1247  tfisi  4580  tfrcllemsucfn  6344  sbthlemi6  6951  sbthlemi8  6953  div32apd  8743  div13apd  8744  expdivapd  10635  modfsummodlemstep  11431  pcqmul  12268  pcid  12288  pcneg  12289  pc2dvds  12294  pcz  12296  pcaddlem  12303  pcadd  12304  pcmpt2  12307  pcbc  12314  qexpz  12315  expnprm  12316  ennnfonelemg  12369  ssblex  13482