MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  syl312anc Structured version   Visualization version   GIF version

Theorem syl312anc 1391
Description: Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
Hypotheses
Ref Expression
syl3anc.1 (𝜑𝜓)
syl3anc.2 (𝜑𝜒)
syl3anc.3 (𝜑𝜃)
syl3Xanc.4 (𝜑𝜏)
syl23anc.5 (𝜑𝜂)
syl33anc.6 (𝜑𝜁)
syl312anc.7 (((𝜓𝜒𝜃) ∧ 𝜏 ∧ (𝜂𝜁)) → 𝜎)
Assertion
Ref Expression
syl312anc (𝜑𝜎)

Proof of Theorem syl312anc
StepHypRef Expression
1 syl3anc.1 . 2 (𝜑𝜓)
2 syl3anc.2 . 2 (𝜑𝜒)
3 syl3anc.3 . 2 (𝜑𝜃)
4 syl3Xanc.4 . 2 (𝜑𝜏)
5 syl23anc.5 . . 3 (𝜑𝜂)
6 syl33anc.6 . . 3 (𝜑𝜁)
75, 6jca 512 . 2 (𝜑 → (𝜂𝜁))
8 syl312anc.7 . 2 (((𝜓𝜒𝜃) ∧ 𝜏 ∧ (𝜂𝜁)) → 𝜎)
91, 2, 3, 4, 7, 8syl311anc 1384 1 (𝜑𝜎)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089
This theorem is referenced by:  pythagtriplem19  16768  cdleme27cl  39323  cdlemefs27cl  39370  cdleme32fvcl  39397  cdlemg16ALTN  39615  cdlemg27a  39649  cdlemg31c  39656  cdlemg39  39673  cdlemk11ta  39886  cdlemk19ylem  39887  cdlemk11tc  39902  cdlemk45  39904  dihmeetlem12N  40275  dihjatc  40374  flt4lem5c  41478  flt4lem5d  41479  flt4lem5e  41480
  Copyright terms: Public domain W3C validator