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Theorem ifval 4570
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.)
Assertion
Ref Expression
ifval (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴 = 𝐵) ∧ (¬ 𝜑𝐴 = 𝐶)))

Proof of Theorem ifval
StepHypRef Expression
1 eqif 4569 . 2 (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴 = 𝐵) ∨ (¬ 𝜑𝐴 = 𝐶)))
2 cases2 1046 . 2 (((𝜑𝐴 = 𝐵) ∨ (¬ 𝜑𝐴 = 𝐶)) ↔ ((𝜑𝐴 = 𝐵) ∧ (¬ 𝜑𝐴 = 𝐶)))
31, 2bitri 274 1 (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴 = 𝐵) ∧ (¬ 𝜑𝐴 = 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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