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| Mirrors > Home > MPE Home > Th. List > syl333anc | Structured version Visualization version GIF version | ||
| Description: A syllogism inference combined with contraction. (Contributed by NM, 10-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) |
| syl333anc.9 | ⊢ (𝜑 → 𝜇) |
| syl333anc.10 | ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁) ∧ (𝜎 ∧ 𝜌 ∧ 𝜇)) → 𝜆) |
| Ref | Expression |
|---|---|
| syl333anc | ⊢ (𝜑 → 𝜆) |
| 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 | syl333anc.9 | . . 3 ⊢ (𝜑 → 𝜇) | |
| 10 | 7, 8, 9 | 3jca 1129 | . 2 ⊢ (𝜑 → (𝜎 ∧ 𝜌 ∧ 𝜇)) |
| 11 | syl333anc.10 | . 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁) ∧ (𝜎 ∧ 𝜌 ∧ 𝜇)) → 𝜆) | |
| 12 | 1, 2, 3, 4, 5, 6, 10, 11 | syl331anc 1397 | 1 ⊢ (𝜑 → 𝜆) |
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