Theorem syl331anc 1398

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 1129	. 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 1387	1 ⊢ (𝜑 → 𝜌)

Syntax hints:   → wi 4   ∧ w3a 1087

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 1089

This theorem is referenced by:  syl332anc  1404  syl333anc  1405  qredeu  16627  brbtwn2  28974  3atlem4  39932  3atlem6  39934  llnexchb2  40315  osumcllem9N  40410  cdlemd4  40647  cdleme26fALTN  40808  cdleme26f  40809  cdleme36m  40907  cdlemg17b  41108  cdlemg17h  41114  cdlemk38  41361  cdlemk53b  41402  cdlemkyyN  41408  cdlemk43N  41409