Theorem syl331anc 1391

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)
syl331anc.8	⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁) ∧ 𝜎) → 𝜌)

Assertion

Ref	Expression
syl331anc	⊢ (𝜑 → 𝜌)

Proof of Theorem syl331anc

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	. . 3 ⊢ (𝜑 → 𝜏)
5		syl23anc.5	. . 3 ⊢ (𝜑 → 𝜂)
6		syl33anc.6	. . 3 ⊢ (𝜑 → 𝜁)
7	4, 5, 6	3jca 1124	. 2 ⊢ (𝜑 → (𝜏 ∧ 𝜂 ∧ 𝜁))
8		syl133anc.7	. 2 ⊢ (𝜑 → 𝜎)
9		syl331anc.8	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁) ∧ 𝜎) → 𝜌)
10	1, 2, 3, 7, 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:  syl332anc  1397  syl333anc  1398  qredeu  16002  brbtwn2  26691  3atlem4  36637  3atlem6  36639  llnexchb2  37020  osumcllem9N  37115  cdlemd4  37352  cdleme26fALTN  37513  cdleme26f  37514  cdleme36m  37612  cdlemg17b  37813  cdlemg17h  37819  cdlemk38  38066  cdlemk53b  38107  cdlemkyyN  38113  cdlemk43N  38114