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Theorem ifval 4532
Description: Another expression of the value of the if predicate, analogous to eqif 4531. See also the more specialized iftrue 4496 and iffalse 4499. (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 4531 . 2 (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴 = 𝐵) ∨ (¬ 𝜑𝐴 = 𝐶)))
2 cases2 1047 . 2 (((𝜑𝐴 = 𝐵) ∨ (¬ 𝜑𝐴 = 𝐶)) ↔ ((𝜑𝐴 = 𝐵) ∧ (¬ 𝜑𝐴 = 𝐶)))
31, 2bitri 275 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 397  wo 846   = wceq 1542  ifcif 4490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-if 4491
This theorem is referenced by:  dfiota4  6492  bj-projval  35517  dfaiota3  45414
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