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Theorem syl212anc 1379
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 1375 1 (𝜑𝜁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086
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 1088
This theorem is referenced by:  pntrmax  26712  tglineineq  27004  tglineinteq  27006  paddasslem4  37837  4atexlemu  38078  4atexlemv  38079  cdleme20aN  38323  cdleme20g  38329  cdlemg9a  38646  cdlemg12a  38657  cdlemg17dALTN  38678  cdlemg18b  38693  cdlemg18c  38694
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