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Mirrors > Home > MPE Home > Th. List > elif | Structured version Visualization version GIF version |
Description: Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.) |
Ref | Expression |
---|---|
elif | ⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 ∈ 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 ∈ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2828 | . 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ 𝐴 ∈ 𝐵)) | |
2 | eleq2 2828 | . 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ 𝐴 ∈ 𝐶)) | |
3 | 1, 2 | elimif 4568 | 1 ⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 ∈ 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 ∈ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∈ wcel 2106 ifcif 4531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-if 4532 |
This theorem is referenced by: bj-imdirco 37173 safesnsupfiss 43405 clsk1indlem3 44033 |
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