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Theorem ifval 4480
Description: Another expression of the value of the if predicate, analogous to eqif 4479. See also the more specialized iftrue 4445 and iffalse 4448. (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 4479 . 2 (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴 = 𝐵) ∨ (¬ 𝜑𝐴 = 𝐶)))
2 cases2 1043 . 2 (((𝜑𝐴 = 𝐵) ∨ (¬ 𝜑𝐴 = 𝐶)) ↔ ((𝜑𝐴 = 𝐵) ∧ (¬ 𝜑𝐴 = 𝐶)))
31, 2bitri 278 1 (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴 = 𝐵) ∧ (¬ 𝜑𝐴 = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  ifcif 4439
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-if 4440
This theorem is referenced by:  dfiota4  6326  bj-projval  34393
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