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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 512 . 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 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:  4atlem11  37623  dalem52  37738  dath2  37751  dalawlem1  37885  dalaw  37900  cdlemb2  38055  4atexlem7  38089  cdleme7ga  38262  cdleme18a  38305  cdleme18c  38307  cdleme21f  38346  cdleme26f2ALTN  38378  cdleme26f2  38379  cdleme27a  38381  cdlemg17dN  38677  cdlemg18a  38692  cdlemg31d  38714  cdlemg48  38751  cdlemj1  38835  dihord4  39272
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