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| Mirrors > Home > MPE Home > Th. List > syl3anc | 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) |
| syl3anc.4 | ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) |
| Ref | Expression |
|---|---|
| syl3anc | ⊢ (𝜑 → 𝜏) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | syl3anc.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
| 2 | syl3anc.2 | . . 3 ⊢ (𝜑 → 𝜒) | |
| 3 | syl3anc.3 | . . 3 ⊢ (𝜑 → 𝜃) | |
| 4 | 1, 2, 3 | 3jca 1129 | . 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃)) |
| 5 | syl3anc.4 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) | |
| 6 | 4, 5 | syl 17 | 1 ⊢ (𝜑 → 𝜏) |
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