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Theorem ifeqor 3699
 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 3668 . . . 4 (φ → if(φ, A, B) = A)
21con3i 127 . . 3 (¬ if(φ, A, B) = A → ¬ φ)
3 iffalse 3669 . . 3 φ → if(φ, A, B) = B)
42, 3syl 15 . 2 (¬ if(φ, A, B) = A → if(φ, A, B) = B)
54orri 365 1 ( if(φ, A, B) = A if(φ, A, B) = B)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 357   = wceq 1642   ifcif 3662 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3663 This theorem is referenced by:  ifpr  3774  enprmaplem5  6080
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