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| Mirrors > Home > MPE Home > Th. List > iffalse | Structured version Visualization version GIF version | ||
| Description: Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.) |
| Ref | Expression |
|---|---|
| iffalse | ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-if 4526 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} | |
| 2 | dedlemb 1047 | . . 3 ⊢ (¬ 𝜑 → (𝑥 ∈ 𝐵 ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)))) | |
| 3 | 2 | eqabdv 2875 | . 2 ⊢ (¬ 𝜑 → 𝐵 = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}) |
| 4 | 1, 3 | eqtr4id 2796 | 1 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) |
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