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Theorem syl332anc 1395
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 512 . 2 (𝜑 → (𝜎𝜌))
10 syl332anc.9 . 2 (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ (𝜎𝜌)) → 𝜇)
111, 2, 3, 4, 5, 6, 9, 10syl331anc 1389 1 (𝜑𝜇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081
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 208  df-an 397  df-3an 1083
This theorem is referenced by:  mdetunilem5  21144  mdetuni0  21149  lnjatN  36786  lncmp  36789  cdlema1N  36797  4atexlemex6  37080  cdlemd4  37207  cdleme18c  37299  cdleme18d  37301  cdleme19b  37310  cdleme21ct  37335  cdleme21d  37336  cdleme21e  37337  cdleme21k  37344  cdleme22g  37354  cdleme24  37358  cdleme27a  37373  cdleme27N  37375  cdleme28a  37376  cdleme40n  37474  cdlemg16zz  37666  cdlemg37  37695  cdlemk21-2N  37897  cdlemk20-2N  37898  cdlemk28-3  37914  cdlemk19xlem  37948
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