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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 | dedlemb 1027 | . . 3 ⊢ (¬ 𝜑 → (𝑥 ∈ 𝐵 ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)))) | |
2 | 1 | abbi2dv 2903 | . 2 ⊢ (¬ 𝜑 → 𝐵 = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}) |
3 | df-if 4351 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} | |
4 | 2, 3 | syl6reqr 2834 | 1 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) |
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