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Theorem bj-df-ifc 34023
Description: Candidate definition for the conditional operator for classes. This is in line with the definition of a class as the extension of a predicate in df-clab 2777. We reprove the current df-if 4426 from it in bj-dfif 34024. (Contributed by BJ, 20-Sep-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-df-ifc if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)}
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵

Proof of Theorem bj-df-ifc
StepHypRef Expression
1 df-if 4426 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
2 ancom 464 . . . . 5 ((𝑥𝐴𝜑) ↔ (𝜑𝑥𝐴))
3 ancom 464 . . . . 5 ((𝑥𝐵 ∧ ¬ 𝜑) ↔ (¬ 𝜑𝑥𝐵))
42, 3orbi12i 912 . . . 4 (((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑)) ↔ ((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵)))
5 df-ifp 1059 . . . 4 (if-(𝜑, 𝑥𝐴, 𝑥𝐵) ↔ ((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵)))
64, 5bitr4i 281 . . 3 (((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑)) ↔ if-(𝜑, 𝑥𝐴, 𝑥𝐵))
76abbii 2863 . 2 {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)}
81, 7eqtri 2821 1 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399  wo 844  if-wif 1058   = wceq 1538  wcel 2111  {cab 2776  ifcif 4425
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 4426
This theorem is referenced by:  bj-dfif  34024  bj-ififc  34025
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