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Theorem ifeqor 4326
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 4283 . . . 4 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
21con3i 152 . . 3 (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → ¬ 𝜑)
32iffalsed 4288 . 2 (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → if(𝜑, 𝐴, 𝐵) = 𝐵)
43orri 889 1 (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 874   = wceq 1653  ifcif 4277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-if 4278
This theorem is referenced by:  ifpr  4423  rabrsn  4448  prmolefac  16083  muval2  25212  finxpreclem2  33725  relexpxpmin  38792
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