Theorem syl121anc 1254

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 1249	1 |- ( ph -> et )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980

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 982

This theorem is referenced by:  syl122anc  1258  tfisi  4604  tfrcllemsucfn  6379  sbthlemi6  6992  sbthlemi8  6994  div32apd  8802  div13apd  8803  expdivapd  10702  modfsummodlemstep  11500  pcqmul  12338  pcid  12359  pcneg  12360  pc2dvds  12365  pcz  12367  pcaddlem  12374  pcadd  12375  pcmpt2  12379  pcbc  12386  qexpz  12387  expnprm  12388  ennnfonelemg  12457  ssblex  14408