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Theorem syl212anc 1486
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
Hypotheses
Ref Expression
syl12anc.1 (𝜑𝜓)
syl12anc.2 (𝜑𝜒)
syl12anc.3 (𝜑𝜃)
syl22anc.4 (𝜑𝜏)
syl23anc.5 (𝜑𝜂)
syl212anc.6 (((𝜓𝜒) ∧ 𝜃 ∧ (𝜏𝜂)) → 𝜁)
Assertion
Ref Expression
syl212anc (𝜑𝜁)

Proof of Theorem syl212anc
StepHypRef Expression
1 syl12anc.1 . 2 (𝜑𝜓)
2 syl12anc.2 . 2 (𝜑𝜒)
3 syl12anc.3 . 2 (𝜑𝜃)
4 syl22anc.4 . . 3 (𝜑𝜏)
5 syl23anc.5 . . 3 (𝜑𝜂)
64, 5jca 501 . 2 (𝜑 → (𝜏𝜂))
7 syl212anc.6 . 2 (((𝜓𝜒) ∧ 𝜃 ∧ (𝜏𝜂)) → 𝜁)
81, 2, 3, 6, 7syl211anc 1482 1 (𝜑𝜁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071
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 197  df-an 383  df-3an 1073
This theorem is referenced by:  pntrmax  25474  tglineineq  25759  tglineinteq  25761  paddasslem4  35632  4atexlemu  35873  4atexlemv  35874  cdleme20aN  36119  cdleme20g  36125  cdlemg9a  36442  cdlemg12a  36453  cdlemg17dALTN  36474  cdlemg18b  36489  cdlemg18c  36490
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