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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 |
---|---|
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) |
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 | syl3anc.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | syl3anc.2 | . 2 ⊢ (𝜑 → 𝜒) | |
3 | syl3anc.3 | . 2 ⊢ (𝜑 → 𝜃) | |
4 | syl3Xanc.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 512 | . 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 1395 | 1 ⊢ (𝜑 → 𝜇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ 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 206 df-an 397 df-3an 1089 |
This theorem is referenced by: mdetunilem5 22125 mdetuni0 22130 lnjatN 38737 lncmp 38740 cdlema1N 38748 4atexlemex6 39031 cdlemd4 39158 cdleme18c 39250 cdleme18d 39252 cdleme19b 39261 cdleme21ct 39286 cdleme21d 39287 cdleme21e 39288 cdleme21k 39295 cdleme22g 39305 cdleme24 39309 cdleme27a 39324 cdleme27N 39326 cdleme28a 39327 cdleme40n 39425 cdlemg16zz 39617 cdlemg37 39646 cdlemk21-2N 39848 cdlemk20-2N 39849 cdlemk28-3 39865 cdlemk19xlem 39899 |
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