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| Mirrors > Home > MPE Home > Th. List > mpbir3and | Structured version Visualization version GIF version | ||
| Description: Detach a conjunction of truths in a biconditional. (Contributed by Mario Carneiro, 11-May-2014.) (Revised by Mario Carneiro, 9-Jan-2015.) |
| Ref | Expression |
|---|---|
| mpbir3and.1 | ⊢ (𝜑 → 𝜒) |
| mpbir3and.2 | ⊢ (𝜑 → 𝜃) |
| mpbir3and.3 | ⊢ (𝜑 → 𝜏) |
| mpbir3and.4 | ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏))) |
| Ref | Expression |
|---|---|
| mpbir3and | ⊢ (𝜑 → 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mpbir3and.1 | . . 3 ⊢ (𝜑 → 𝜒) | |
| 2 | mpbir3and.2 | . . 3 ⊢ (𝜑 → 𝜃) | |
| 3 | mpbir3and.3 | . . 3 ⊢ (𝜑 → 𝜏) | |
| 4 | 1, 2, 3 | 3jca 1129 | . 2 ⊢ (𝜑 → (𝜒 ∧ 𝜃 ∧ 𝜏)) |
| 5 | mpbir3and.4 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏))) | |
| 6 | 4, 5 | mpbird 257 | 1 ⊢ (𝜑 → 𝜓) |
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