Theorem syl332anc 1424

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 520	. 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 1418	1 ⊢ (𝜑 → 𝜇)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101

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 210  df-an 401  df-3an 1103

This theorem is referenced by:  mdetunilem5  22734  mdetuni0  22739  lnjatN  40416  lncmp  40419  cdlema1N  40427  4atexlemex6  40710  cdlemd4  40837  cdleme18c  40929  cdleme18d  40931  cdleme19b  40940  cdleme21ct  40965  cdleme21d  40966  cdleme21e  40967  cdleme21k  40974  cdleme22g  40984  cdleme24  40988  cdleme27a  41003  cdleme27N  41005  cdleme28a  41006  cdleme40n  41104  cdlemg16zz  41296  cdlemg37  41325  cdlemk21-2N  41527  cdlemk20-2N  41528  cdlemk28-3  41544  cdlemk19xlem  41578