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| Mirrors > Home > MPE Home > Th. List > ifbid | Structured version Visualization version GIF version | ||
| Description: Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.) |
| Ref | Expression |
|---|---|
| ifbid.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| ifbid | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ifbid.1 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 2 | ifbi 4548 | . 2 ⊢ ((𝜓 ↔ 𝜒) → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵)) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵)) |
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