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| Mirrors > Home > MPE Home > Th. List > simplbiim | Structured version Visualization version GIF version | ||
| Description: Implication from an eliminated conjunct equivalent to the antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Wolf Lammen, 26-Mar-2022.) |
| Ref | Expression |
|---|---|
| simplbiim.1 | ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) |
| simplbiim.2 | ⊢ (𝜒 → 𝜃) |
| Ref | Expression |
|---|---|
| simplbiim | ⊢ (𝜑 → 𝜃) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simplbiim.1 | . . 3 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) | |
| 2 | 1 | simprbi 496 | . 2 ⊢ (𝜑 → 𝜒) |
| 3 | simplbiim.2 | . 2 ⊢ (𝜒 → 𝜃) | |
| 4 | 2, 3 | syl 17 | 1 ⊢ (𝜑 → 𝜃) |
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