Theorem syl331anc 1397

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 1128	. 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 1386	1 ⊢ (𝜑 → 𝜌)

Syntax hints:   → wi 4   ∧ 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:  syl332anc  1403  syl333anc  1404  qredeu  16677  brbtwn2  28884  3atlem4  39505  3atlem6  39507  llnexchb2  39888  osumcllem9N  39983  cdlemd4  40220  cdleme26fALTN  40381  cdleme26f  40382  cdleme36m  40480  cdlemg17b  40681  cdlemg17h  40687  cdlemk38  40934  cdlemk53b  40975  cdlemkyyN  40981  cdlemk43N  40982