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Mirrors > Home > MPE Home > Th. List > syl332anc | Structured version Visualization version GIF version |
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) |
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) |
syl233anc.8 | ⊢ (𝜑 → 𝜌) |
syl332anc.9 | ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁) ∧ (𝜎 ∧ 𝜌)) → 𝜇) |
Ref | Expression |
---|---|
syl332anc | ⊢ (𝜑 → 𝜇) |
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 | . 2 ⊢ (𝜑 → 𝜏) | |
5 | syl23anc.5 | . 2 ⊢ (𝜑 → 𝜂) | |
6 | syl33anc.6 | . 2 ⊢ (𝜑 → 𝜁) | |
7 | syl133anc.7 | . . 3 ⊢ (𝜑 → 𝜎) | |
8 | syl233anc.8 | . . 3 ⊢ (𝜑 → 𝜌) | |
9 | 7, 8 | jca 553 | . 2 ⊢ (𝜑 → (𝜎 ∧ 𝜌)) |
10 | syl332anc.9 | . 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁) ∧ (𝜎 ∧ 𝜌)) → 𝜇) | |
11 | 1, 2, 3, 4, 5, 6, 9, 10 | syl331anc 1343 | 1 ⊢ (𝜑 → 𝜇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ 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: mdetunilem5 20241 mdetuni0 20246 lnjatN 34084 lncmp 34087 cdlema1N 34095 4atexlemex6 34378 cdlemd4 34506 cdleme18c 34598 cdleme18d 34600 cdleme19b 34610 cdleme21ct 34635 cdleme21d 34636 cdleme21e 34637 cdleme21k 34644 cdleme22g 34654 cdleme24 34658 cdleme27a 34673 cdleme27N 34675 cdleme28a 34676 cdleme40n 34774 cdlemg16zz 34966 cdlemg37 34995 cdlemk21-2N 35197 cdlemk20-2N 35198 cdlemk28-3 35214 cdlemk19xlem 35248 |
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