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Mirrors > Home > MPE Home > Th. List > ifval | Structured version Visualization version GIF version |
Description: Another expression of the value of the if predicate, analogous to eqif 4569. See also the more specialized iftrue 4534 and iffalse 4537. (Contributed by BJ, 6-Apr-2019.) |
Ref | Expression |
---|---|
ifval | ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 → 𝐴 = 𝐵) ∧ (¬ 𝜑 → 𝐴 = 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqif 4569 | . 2 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶))) | |
2 | cases2 1046 | . 2 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶)) ↔ ((𝜑 → 𝐴 = 𝐵) ∧ (¬ 𝜑 → 𝐴 = 𝐶))) | |
3 | 1, 2 | bitri 274 | 1 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 → 𝐴 = 𝐵) ∧ (¬ 𝜑 → 𝐴 = 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 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: dfiota4 6535 bj-projval 35872 dfaiota3 45790 |
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