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  4635  tfrcllemsucfn  6439  sbthlemi6  7064  sbthlemi8  7066  div32apd  8887  div13apd  8888  expdivapd  10832  swrdsbslen  11119  modfsummodlemstep  11768  pcqmul  12626  pcid  12647  pcneg  12648  pc2dvds  12653  pcz  12655  pcaddlem  12662  pcadd  12663  pcmpt2  12667  pcbc  12674  qexpz  12675  expnprm  12676  ennnfonelemg  12774  ssblex  14903