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Theorem eqif 4571
Description: Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005.)
Assertion
Ref Expression
eqif (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴 = 𝐵) ∨ (¬ 𝜑𝐴 = 𝐶)))

Proof of Theorem eqif
StepHypRef Expression
1 eqeq2 2746 . 2 (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ 𝐴 = 𝐵))
2 eqeq2 2746 . 2 (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ 𝐴 = 𝐶))
31, 2elimif 4567 1 (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴 = 𝐵) ∨ (¬ 𝜑𝐴 = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wo 847   = wceq 1536  ifcif 4530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-if 4531
This theorem is referenced by:  ifval  4572  xpima  6203  fin23lem19  10373  fin23lem28  10377  fin23lem29  10378  fin23lem30  10379  aalioulem3  26390  ifnebib  32569  iocinif  32789  fsumcvg4  33910  ind1a  33999  esumsnf  34044  itg2addnclem2  37658  clsk1indlem4  44033  afvpcfv0  47095
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