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Theorem bj-df-ifc 35457
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 2711. We reprove the current df-if 4530 from it in bj-dfif 35458. (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 4530 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
2 ancom 462 . . . . 5 ((𝑥𝐴𝜑) ↔ (𝜑𝑥𝐴))
3 ancom 462 . . . . 5 ((𝑥𝐵 ∧ ¬ 𝜑) ↔ (¬ 𝜑𝑥𝐵))
42, 3orbi12i 914 . . . 4 (((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑)) ↔ ((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵)))
5 df-ifp 1063 . . . 4 (if-(𝜑, 𝑥𝐴, 𝑥𝐵) ↔ ((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵)))
64, 5bitr4i 278 . . 3 (((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑)) ↔ if-(𝜑, 𝑥𝐴, 𝑥𝐵))
76abbii 2803 . 2 {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)}
81, 7eqtri 2761 1 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397  wo 846  if-wif 1062   = wceq 1542  wcel 2107  {cab 2710  ifcif 4529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-if 4530
This theorem is referenced by:  bj-dfif  35458  bj-ififc  35459
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