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| Mirrors > Home > MPE Home > Th. List > ifbieq2d | Structured version Visualization version GIF version | ||
| Description: Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
| Ref | Expression |
|---|---|
| ifbieq2d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
| ifbieq2d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| Ref | Expression |
|---|---|
| ifbieq2d | ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ifbieq2d.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 2 | 1 | ifbid 4549 | . 2 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐴)) |
| 3 | ifbieq2d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 4 | 3 | ifeq2d 4546 | . 2 ⊢ (𝜑 → if(𝜒, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵)) |
| 5 | 2, 4 | eqtrd 2777 | 1 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵)) |
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