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Theorem bj-df-ifc 32690
Description: The definition of "ifc" if "if-" enters the main part. This is in line with the definition of a class as the extension of a predicate in df-clab 2638. (Contributed by BJ, 20-Sep-2019.)
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 bj-dfifc2 32689 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵))}
2 df-ifp 1033 . . . 4 (if-(𝜑, 𝑥𝐴, 𝑥𝐵) ↔ ((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵)))
32bicomi 214 . . 3 (((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵)) ↔ if-(𝜑, 𝑥𝐴, 𝑥𝐵))
43abbii 2768 . 2 {𝑥 ∣ ((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵))} = {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)}
51, 4eqtri 2673 1 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 382  wa 383  if-wif 1032   = wceq 1523  wcel 2030  {cab 2637  ifcif 4119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-if 4120
This theorem is referenced by:  bj-ififc  32691
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