Theorem syl331anc 1343

Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)

Hypotheses

Ref	Expression
syl12anc.1	⊢ (𝜑 → 𝜓)
syl12anc.2	⊢ (𝜑 → 𝜒)
syl12anc.3	⊢ (𝜑 → 𝜃)
syl22anc.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		syl12anc.1	. 2 ⊢ (𝜑 → 𝜓)
2		syl12anc.2	. 2 ⊢ (𝜑 → 𝜒)
3		syl12anc.3	. 2 ⊢ (𝜑 → 𝜃)
4		syl22anc.4	. . 3 ⊢ (𝜑 → 𝜏)
5		syl23anc.5	. . 3 ⊢ (𝜑 → 𝜂)
6		syl33anc.6	. . 3 ⊢ (𝜑 → 𝜁)
7	4, 5, 6	3jca 1235	. 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 1332	1 ⊢ (𝜑 → 𝜌)

Syntax hints:   → wi 4   ∧ w3a 1031

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 196  df-an 385  df-3an 1033

This theorem is referenced by:  syl332anc  1349  syl333anc  1350  qredeu  15210  brbtwn2  25585  3atlem4  33790  3atlem6  33792  llnexchb2  34173  osumcllem9N  34268  cdlemd4  34506  cdleme26fALTN  34668  cdleme26f  34669  cdleme36m  34767  cdlemg17b  34968  cdlemg17h  34974  cdlemk38  35221  cdlemk53b  35262  cdlemkyyN  35268  cdlemk43N  35269