Theorem syl313anc 1390

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)
syl313anc.8	⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏 ∧ (𝜂 ∧ 𝜁 ∧ 𝜎)) → 𝜌)

Assertion

Ref	Expression
syl313anc	⊢ (𝜑 → 𝜌)

Proof of Theorem syl313anc

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	. . 3 ⊢ (𝜑 → 𝜂)
6		syl33anc.6	. . 3 ⊢ (𝜑 → 𝜁)
7		syl133anc.7	. . 3 ⊢ (𝜑 → 𝜎)
8	5, 6, 7	3jca 1124	. 2 ⊢ (𝜑 → (𝜂 ∧ 𝜁 ∧ 𝜎))
9		syl313anc.8	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏 ∧ (𝜂 ∧ 𝜁 ∧ 𝜎)) → 𝜌)
10	1, 2, 3, 4, 8, 9	syl311anc 1380	1 ⊢ (𝜑 → 𝜌)

Syntax hints:   → wi 4   ∧ w3a 1083

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 209  df-an 399  df-3an 1085

This theorem is referenced by:  syl323anc  1396  osumcllem6N  37091  cdlemg13  37782  cdlemk7u  38000  cdlemk31  38026  cdlemk27-3  38037  cdlemk19ylem  38060  cdlemk46  38078