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| Mirrors > Home > MPE Home > Th. List > syl231anc | 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) | 
| syl231anc.7 | ⊢ (((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏 ∧ 𝜂) ∧ 𝜁) → 𝜎) | 
| Ref | Expression | 
|---|---|
| syl231anc | ⊢ (𝜑 → 𝜎) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | syl3anc.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
| 2 | syl3anc.2 | . . 3 ⊢ (𝜑 → 𝜒) | |
| 3 | 1, 2 | jca 511 | . 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒)) | 
| 4 | syl3anc.3 | . 2 ⊢ (𝜑 → 𝜃) | |
| 5 | syl3Xanc.4 | . 2 ⊢ (𝜑 → 𝜏) | |
| 6 | syl23anc.5 | . 2 ⊢ (𝜑 → 𝜂) | |
| 7 | syl33anc.6 | . 2 ⊢ (𝜑 → 𝜁) | |
| 8 | syl231anc.7 | . 2 ⊢ (((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏 ∧ 𝜂) ∧ 𝜁) → 𝜎) | |
| 9 | 3, 4, 5, 6, 7, 8 | syl131anc 1385 | 1 ⊢ (𝜑 → 𝜎) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ 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 207 df-an 396 df-3an 1089 | 
| This theorem is referenced by: syl232anc 1399 isosctr 26864 axeuclid 28978 dalawlem3 39875 dalawlem6 39878 cdlemd7 40206 cdleme18c 40295 cdlemi 40822 cdlemk7 40850 cdlemk11 40851 cdlemk7u 40872 cdlemk11u 40873 cdlemk19xlem 40944 cdlemk55u1 40967 cdlemk56 40973 | 
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