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Mirrors > Home > MPE Home > Th. List > ifel | Structured version Visualization version GIF version |
Description: Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.) |
Ref | Expression |
---|---|
ifel | ⊢ (if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ ((𝜑 ∧ 𝐴 ∈ 𝐶) ∨ (¬ 𝜑 ∧ 𝐵 ∈ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2900 | . 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) | |
2 | eleq1 2900 | . 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
3 | 1, 2 | elimif 4502 | 1 ⊢ (if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ ((𝜑 ∧ 𝐴 ∈ 𝐶) ∨ (¬ 𝜑 ∧ 𝐵 ∈ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∈ wcel 2110 ifcif 4466 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-if 4467 |
This theorem is referenced by: clwlkclwwlklem2a 27770 smatrcl 31056 clsk1independent 40389 |
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