Theorem syl233anc 1396

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)
syl233anc.9	⊢ (((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏 ∧ 𝜂) ∧ (𝜁 ∧ 𝜎 ∧ 𝜌)) → 𝜇)

Assertion

Ref	Expression
syl233anc	⊢ (𝜑 → 𝜇)

Proof of Theorem syl233anc

Step	Hyp	Ref	Expression
1		syl3anc.1	. . 3 ⊢ (𝜑 → 𝜓)
2		syl3anc.2	. . 3 ⊢ (𝜑 → 𝜒)
3	1, 2	jca 515	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒))
4		syl3anc.3	. 2 ⊢ (𝜑 → 𝜃)
5		syl3Xanc.4	. 2 ⊢ (𝜑 → 𝜏)
6		syl23anc.5	. 2 ⊢ (𝜑 → 𝜂)
7		syl33anc.6	. 2 ⊢ (𝜑 → 𝜁)
8		syl133anc.7	. 2 ⊢ (𝜑 → 𝜎)
9		syl233anc.8	. 2 ⊢ (𝜑 → 𝜌)
10		syl233anc.9	. 2 ⊢ (((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏 ∧ 𝜂) ∧ (𝜁 ∧ 𝜎 ∧ 𝜌)) → 𝜇)
11	3, 4, 5, 6, 7, 8, 9, 10	syl133anc 1390	1 ⊢ (𝜑 → 𝜇)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084

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 400  df-3an 1086

This theorem is referenced by:  br8d  30374  2llnjN  36863  cdleme16b  37575  cdleme18d  37591  cdleme19d  37602  cdleme20bN  37606  cdleme20l1  37616  cdleme22cN  37638  cdleme22eALTN  37641  cdleme22f  37642  cdlemg33c0  37998  cdlemk5  38132  cdlemk5u  38157  cdlemky  38222  cdlemkyyN  38258