MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  syl212anc Structured version   Visualization version   GIF version

Theorem syl212anc 1380
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
Hypotheses
Ref Expression
syl3anc.1 (𝜑𝜓)
syl3anc.2 (𝜑𝜒)
syl3anc.3 (𝜑𝜃)
syl3Xanc.4 (𝜑𝜏)
syl23anc.5 (𝜑𝜂)
syl212anc.6 (((𝜓𝜒) ∧ 𝜃 ∧ (𝜏𝜂)) → 𝜁)
Assertion
Ref Expression
syl212anc (𝜑𝜁)

Proof of Theorem syl212anc
StepHypRef Expression
1 syl3anc.1 . 2 (𝜑𝜓)
2 syl3anc.2 . 2 (𝜑𝜒)
3 syl3anc.3 . 2 (𝜑𝜃)
4 syl3Xanc.4 . . 3 (𝜑𝜏)
5 syl23anc.5 . . 3 (𝜑𝜂)
64, 5jca 512 . 2 (𝜑 → (𝜏𝜂))
7 syl212anc.6 . 2 (((𝜓𝜒) ∧ 𝜃 ∧ (𝜏𝜂)) → 𝜁)
81, 2, 3, 6, 7syl211anc 1376 1 (𝜑𝜁)
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:  pntrmax  26908  tglineineq  27483  tglineinteq  27485  paddasslem4  38275  4atexlemu  38516  4atexlemv  38517  cdleme20aN  38761  cdleme20g  38767  cdlemg9a  39084  cdlemg12a  39095  cdlemg17dALTN  39116  cdlemg18b  39131  cdlemg18c  39132
  Copyright terms: Public domain W3C validator