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| Mirrors > Home > MPE Home > Th. List > iftrue | Structured version Visualization version GIF version | ||
| Description: Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
| Ref | Expression |
|---|---|
| iftrue | ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfif2 4527 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑))} | |
| 2 | dedlem0a 1044 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ ((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑)))) | |
| 3 | 2 | eqabdv 2875 | . 2 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ ((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑))}) |
| 4 | 1, 3 | eqtr4id 2796 | 1 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) |
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