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| Mirrors > Home > MPE Home > Th. List > ifbieq12d | Structured version Visualization version GIF version | ||
| Description: Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.) |
| Ref | Expression |
|---|---|
| ifbieq12d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
| ifbieq12d.2 | ⊢ (𝜑 → 𝐴 = 𝐶) |
| ifbieq12d.3 | ⊢ (𝜑 → 𝐵 = 𝐷) |
| Ref | Expression |
|---|---|
| ifbieq12d | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ifbieq12d.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 2 | 1 | ifbid 4549 | . 2 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵)) |
| 3 | ifbieq12d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐶) | |
| 4 | ifbieq12d.3 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐷) | |
| 5 | 3, 4 | ifeq12d 4547 | . 2 ⊢ (𝜑 → if(𝜒, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷)) |
| 6 | 2, 5 | eqtrd 2777 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷)) |
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