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  22529  mdetuni0  22534  lnjatN  39818  lncmp  39821  cdlema1N  39829  4atexlemex6  40112  cdlemd4  40239  cdleme18c  40331  cdleme18d  40333  cdleme19b  40342  cdleme21ct  40367  cdleme21d  40368  cdleme21e  40369  cdleme21k  40376  cdleme22g  40386  cdleme24  40390  cdleme27a  40405  cdleme27N  40407  cdleme28a  40408  cdleme40n  40506  cdlemg16zz  40698  cdlemg37  40727  cdlemk21-2N  40929  cdlemk20-2N  40930  cdlemk28-3  40946  cdlemk19xlem  40980