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Theorem syl332anc 1400
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 511 . 2 (𝜑 → (𝜎𝜌))
10 syl332anc.9 . 2 (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ (𝜎𝜌)) → 𝜇)
111, 2, 3, 4, 5, 6, 9, 10syl331anc 1394 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 207  df-an 396  df-3an 1088
This theorem is referenced by:  mdetunilem5  22638  mdetuni0  22643  lnjatN  39763  lncmp  39766  cdlema1N  39774  4atexlemex6  40057  cdlemd4  40184  cdleme18c  40276  cdleme18d  40278  cdleme19b  40287  cdleme21ct  40312  cdleme21d  40313  cdleme21e  40314  cdleme21k  40321  cdleme22g  40331  cdleme24  40335  cdleme27a  40350  cdleme27N  40352  cdleme28a  40353  cdleme40n  40451  cdlemg16zz  40643  cdlemg37  40672  cdlemk21-2N  40874  cdlemk20-2N  40875  cdlemk28-3  40891  cdlemk19xlem  40925
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