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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 22109 mdetuni0 22114 lnjatN 38639 lncmp 38642 cdlema1N 38650 4atexlemex6 38933 cdlemd4 39060 cdleme18c 39152 cdleme18d 39154 cdleme19b 39163 cdleme21ct 39188 cdleme21d 39189 cdleme21e 39190 cdleme21k 39197 cdleme22g 39207 cdleme24 39211 cdleme27a 39226 cdleme27N 39228 cdleme28a 39229 cdleme40n 39327 cdlemg16zz 39519 cdlemg37 39548 cdlemk21-2N 39750 cdlemk20-2N 39751 cdlemk28-3 39767 cdlemk19xlem 39801 |
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