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| Mirrors > Home > MPE Home > Th. List > mpbir2and | Structured version Visualization version GIF version | ||
| Description: Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2012.) |
| Ref | Expression |
|---|---|
| mpbir2and.1 | ⊢ (𝜑 → 𝜒) |
| mpbir2and.2 | ⊢ (𝜑 → 𝜃) |
| mpbir2and.3 | ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) |
| Ref | Expression |
|---|---|
| mpbir2and | ⊢ (𝜑 → 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mpbir2and.1 | . . 3 ⊢ (𝜑 → 𝜒) | |
| 2 | mpbir2and.2 | . . 3 ⊢ (𝜑 → 𝜃) | |
| 3 | 1, 2 | jca 511 | . 2 ⊢ (𝜑 → (𝜒 ∧ 𝜃)) |
| 4 | mpbir2and.3 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) | |
| 5 | 3, 4 | mpbird 257 | 1 ⊢ (𝜑 → 𝜓) |
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