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Theorem ifeqor 4541
Description: The possible values of a conditional operator. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
ifeqor (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵)

Proof of Theorem ifeqor
StepHypRef Expression
1 iftrue 4496 . . . 4 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
21con3i 154 . . 3 (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → ¬ 𝜑)
32iffalsed 4501 . 2 (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → if(𝜑, 𝐴, 𝐵) = 𝐵)
43orri 861 1 (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  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:  ifpr  4656  rabrsn  4689  prmolefac  16926  muval2  26506  finxpreclem2  35911  relexpxpmin  42081
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