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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  16705  cdleme27cl  38829  cdlemefs27cl  38876  cdleme32fvcl  38903  cdlemg16ALTN  39121  cdlemg27a  39155  cdlemg31c  39162  cdlemg39  39179  cdlemk11ta  39392  cdlemk19ylem  39393  cdlemk11tc  39408  cdlemk45  39410  dihmeetlem12N  39781  dihjatc  39880  flt4lem5c  40978  flt4lem5d  40979  flt4lem5e  40980
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