Theorem syl332anc 1401

Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)

Hypotheses

Ref	Expression
syl3anc.1	⊢ (𝜑 → 𝜓)
syl3anc.2	⊢ (𝜑 → 𝜒)
syl3anc.3	⊢ (𝜑 → 𝜃)
syl3Xanc.4	⊢ (𝜑 → 𝜏)
syl23anc.5	⊢ (𝜑 → 𝜂)
syl33anc.6	⊢ (𝜑 → 𝜁)
syl133anc.7	⊢ (𝜑 → 𝜎)
syl233anc.8	⊢ (𝜑 → 𝜌)
syl332anc.9	⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁) ∧ (𝜎 ∧ 𝜌)) → 𝜇)

Assertion

Ref	Expression
syl332anc	⊢ (𝜑 → 𝜇)

Proof of Theorem syl332anc

Step	Hyp	Ref	Expression
1		syl3anc.1	. 2 ⊢ (𝜑 → 𝜓)
2		syl3anc.2	. 2 ⊢ (𝜑 → 𝜒)
3		syl3anc.3	. 2 ⊢ (𝜑 → 𝜃)
4		syl3Xanc.4	. 2 ⊢ (𝜑 → 𝜏)
5		syl23anc.5	. 2 ⊢ (𝜑 → 𝜂)
6		syl33anc.6	. 2 ⊢ (𝜑 → 𝜁)
7		syl133anc.7	. . 3 ⊢ (𝜑 → 𝜎)
8		syl233anc.8	. . 3 ⊢ (𝜑 → 𝜌)
9	7, 8	jca 512	. 2 ⊢ (𝜑 → (𝜎 ∧ 𝜌))
10		syl332anc.9	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁) ∧ (𝜎 ∧ 𝜌)) → 𝜇)
11	1, 2, 3, 4, 5, 6, 9, 10	syl331anc 1395	1 ⊢ (𝜑 → 𝜇)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087

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 1089

This theorem is referenced by:  mdetunilem5  22109  mdetuni0  22114  lnjatN  38639  lncmp  38642  cdlema1N  38650  4atexlemex6  38933  cdlemd4  39060  cdleme18c  39152  cdleme18d  39154  cdleme19b  39163  cdleme21ct  39188  cdleme21d  39189  cdleme21e  39190  cdleme21k  39197  cdleme22g  39207  cdleme24  39211  cdleme27a  39226  cdleme27N  39228  cdleme28a  39229  cdleme40n  39327  cdlemg16zz  39519  cdlemg37  39548  cdlemk21-2N  39750  cdlemk20-2N  39751  cdlemk28-3  39767  cdlemk19xlem  39801