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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 2821 | . 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) | |
2 | eleq1 2821 | . 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
3 | 1, 2 | elimif 4565 | 1 ⊢ (if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ ((𝜑 ∧ 𝐴 ∈ 𝐶) ∨ (¬ 𝜑 ∧ 𝐵 ∈ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∈ wcel 2106 ifcif 4528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-if 4529 |
This theorem is referenced by: clwlkclwwlklem2a 29248 smatrcl 32771 clsk1independent 42787 |
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