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Theorem bj-dfif 34178
 Description: Alternate definition of the conditional operator for classes, which used to be the main definition. (Contributed by BJ, 26-Dec-2023.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-dfif if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵))}
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵

Proof of Theorem bj-dfif
StepHypRef Expression
1 bj-df-ifc 34177 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)}
2 df-ifp 1059 . . 3 (if-(𝜑, 𝑥𝐴, 𝑥𝐵) ↔ ((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵)))
32abbii 2863 . 2 {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)} = {𝑥 ∣ ((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵))}
41, 3eqtri 2821 1 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵))}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 399   ∨ wo 844  if-wif 1058   = wceq 1538   ∈ wcel 2111  {cab 2776  ifcif 4428 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-if 4429 This theorem is referenced by: (None)
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