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Theorem syl332anc 1402
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)
syl332anc.9 (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ (𝜎𝜌)) → 𝜇)
Assertion
Ref Expression
syl332anc (𝜑𝜇)

Proof of Theorem syl332anc
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 . 2 (𝜑𝜂)
6 syl33anc.6 . 2 (𝜑𝜁)
7 syl133anc.7 . . 3 (𝜑𝜎)
8 syl233anc.8 . . 3 (𝜑𝜌)
97, 8jca 513 . 2 (𝜑 → (𝜎𝜌))
10 syl332anc.9 . 2 (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ (𝜎𝜌)) → 𝜇)
111, 2, 3, 4, 5, 6, 9, 10syl331anc 1396 1 (𝜑𝜇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088
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 398  df-3an 1090
This theorem is referenced by:  mdetunilem5  22118  mdetuni0  22123  lnjatN  38651  lncmp  38654  cdlema1N  38662  4atexlemex6  38945  cdlemd4  39072  cdleme18c  39164  cdleme18d  39166  cdleme19b  39175  cdleme21ct  39200  cdleme21d  39201  cdleme21e  39202  cdleme21k  39209  cdleme22g  39219  cdleme24  39223  cdleme27a  39238  cdleme27N  39240  cdleme28a  39241  cdleme40n  39339  cdlemg16zz  39531  cdlemg37  39560  cdlemk21-2N  39762  cdlemk20-2N  39763  cdlemk28-3  39779  cdlemk19xlem  39813
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