Theorem syl331anc 1351

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 1242	. 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 1340	1 ⊢ (𝜑 → 𝜌)

Syntax hints:   → wi 4   ∧ w3a 1037

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 197  df-an 386  df-3an 1039

This theorem is referenced by:  syl332anc  1357  syl333anc  1358  qredeu  15372  brbtwn2  25785  3atlem4  34772  3atlem6  34774  llnexchb2  35155  osumcllem9N  35250  cdlemd4  35488  cdleme26fALTN  35650  cdleme26f  35651  cdleme36m  35749  cdlemg17b  35950  cdlemg17h  35956  cdlemk38  36203  cdlemk53b  36244  cdlemkyyN  36250  cdlemk43N  36251