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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 511 | . 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 1394 | 1 ⊢ (𝜑 → 𝜇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 |
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 207 df-an 396 df-3an 1088 |
This theorem is referenced by: mdetunilem5 22638 mdetuni0 22643 lnjatN 39763 lncmp 39766 cdlema1N 39774 4atexlemex6 40057 cdlemd4 40184 cdleme18c 40276 cdleme18d 40278 cdleme19b 40287 cdleme21ct 40312 cdleme21d 40313 cdleme21e 40314 cdleme21k 40321 cdleme22g 40331 cdleme24 40335 cdleme27a 40350 cdleme27N 40352 cdleme28a 40353 cdleme40n 40451 cdlemg16zz 40643 cdlemg37 40672 cdlemk21-2N 40874 cdlemk20-2N 40875 cdlemk28-3 40891 cdlemk19xlem 40925 |
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