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Theorem syl323anc 1399
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)
syl33anc.6 (𝜑𝜁)
syl133anc.7 (𝜑𝜎)
syl233anc.8 (𝜑𝜌)
syl323anc.9 (((𝜓𝜒𝜃) ∧ (𝜏𝜂) ∧ (𝜁𝜎𝜌)) → 𝜇)
Assertion
Ref Expression
syl323anc (𝜑𝜇)

Proof of Theorem syl323anc
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 511 . 2 (𝜑 → (𝜏𝜂))
7 syl33anc.6 . 2 (𝜑𝜁)
8 syl133anc.7 . 2 (𝜑𝜎)
9 syl233anc.8 . 2 (𝜑𝜌)
10 syl323anc.9 . 2 (((𝜓𝜒𝜃) ∧ (𝜏𝜂) ∧ (𝜁𝜎𝜌)) → 𝜇)
111, 2, 3, 6, 7, 8, 9, 10syl313anc 1393 1 (𝜑𝜇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  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 396  df-3an 1088
This theorem is referenced by:  4atlem11  38784  dalem52  38899  dath2  38912  dalawlem1  39046  dalaw  39061  cdlemb2  39216  4atexlem7  39250  cdleme7ga  39423  cdleme18a  39466  cdleme18c  39468  cdleme21f  39507  cdleme26f2ALTN  39539  cdleme26f2  39540  cdleme27a  39542  cdlemg17dN  39838  cdlemg18a  39853  cdlemg31d  39875  cdlemg48  39912  cdlemj1  39996  dihord4  40433
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