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| Mirrors > Home > MPE Home > Th. List > mpbir3an | Structured version Visualization version GIF version | ||
| Description: Detach a conjunction of truths in a biconditional. (Contributed by NM, 16-Sep-2011.) |
| Ref | Expression |
|---|---|
| mpbir3an.1 | ⊢ 𝜓 |
| mpbir3an.2 | ⊢ 𝜒 |
| mpbir3an.3 | ⊢ 𝜃 |
| mpbir3an.4 | ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) |
| Ref | Expression |
|---|---|
| mpbir3an | ⊢ 𝜑 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mpbir3an.1 | . . 3 ⊢ 𝜓 | |
| 2 | mpbir3an.2 | . . 3 ⊢ 𝜒 | |
| 3 | mpbir3an.3 | . . 3 ⊢ 𝜃 | |
| 4 | 1, 2, 3 | 3pm3.2i 1340 | . 2 ⊢ (𝜓 ∧ 𝜒 ∧ 𝜃) |
| 5 | mpbir3an.4 | . 2 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) | |
| 6 | 4, 5 | mpbir 231 | 1 ⊢ 𝜑 |
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