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Theorem ifeqor 4110
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 4070 . . . 4 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
21con3i 150 . . 3 (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → ¬ 𝜑)
32iffalsed 4075 . 2 (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → if(𝜑, 𝐴, 𝐵) = 𝐵)
43orri 391 1 (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 383   = wceq 1480  ifcif 4064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-if 4065
This theorem is referenced by:  ifpr  4211  rabrsn  4236  prmolefac  15693  muval2  24794  finxpreclem2  32898  relexpxpmin  37529
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