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Theorem syl212anc 1499
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 507 . 2 (𝜑 → (𝜏𝜂))
7 syl212anc.6 . 2 (((𝜓𝜒) ∧ 𝜃 ∧ (𝜏𝜂)) → 𝜁)
81, 2, 3, 6, 7syl211anc 1495 1 (𝜑𝜁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107
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 198  df-an 385  df-3an 1109
This theorem is referenced by:  pntrmax  25544  tglineineq  25829  tglineinteq  25831  paddasslem4  35711  4atexlemu  35952  4atexlemv  35953  cdleme20aN  36197  cdleme20g  36203  cdlemg9a  36520  cdlemg12a  36531  cdlemg17dALTN  36552  cdlemg18b  36567  cdlemg18c  36568
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