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Theorem ifeqor 4277
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 4237 . . . 4 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
21con3i 150 . . 3 (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → ¬ 𝜑)
32iffalsed 4242 . 2 (¬ if(𝜑, 𝐴, 𝐵) = 𝐴 → if(𝜑, 𝐴, 𝐵) = 𝐵)
43orri 390 1 (if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 382   = wceq 1632  ifcif 4231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-if 4232
This theorem is referenced by:  ifpr  4378  rabrsn  4404  prmolefac  15973  muval2  25081  finxpreclem2  33557  relexpxpmin  38530
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