Theorem syl332anc 1403

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 511	. 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 1397	1 ⊢ (𝜑 → 𝜇)

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  22559  mdetuni0  22564  lnjatN  39804  lncmp  39807  cdlema1N  39815  4atexlemex6  40098  cdlemd4  40225  cdleme18c  40317  cdleme18d  40319  cdleme19b  40328  cdleme21ct  40353  cdleme21d  40354  cdleme21e  40355  cdleme21k  40362  cdleme22g  40372  cdleme24  40376  cdleme27a  40391  cdleme27N  40393  cdleme28a  40394  cdleme40n  40492  cdlemg16zz  40684  cdlemg37  40713  cdlemk21-2N  40915  cdlemk20-2N  40916  cdlemk28-3  40932  cdlemk19xlem  40966