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Theorem bj-dfifc2 32206
 Description: This should be the alternate definition of "ifc" if "if-" enters the main part. (Contributed by BJ, 20-Sep-2019.)
Assertion
Ref Expression
bj-dfifc2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵))}
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵

Proof of Theorem bj-dfifc2
StepHypRef Expression
1 df-if 4059 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
2 ancom 466 . . . . 5 ((𝜑𝑥𝐴) ↔ (𝑥𝐴𝜑))
3 ancom 466 . . . . 5 ((¬ 𝜑𝑥𝐵) ↔ (𝑥𝐵 ∧ ¬ 𝜑))
42, 3orbi12i 543 . . . 4 (((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵)) ↔ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑)))
54bicomi 214 . . 3 (((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑)) ↔ ((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵)))
65abbii 2736 . 2 {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ ((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵))}
71, 6eqtri 2643 1 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵))}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 383   ∧ wa 384   = wceq 1480   ∈ wcel 1987  {cab 2607  ifcif 4058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-if 4059 This theorem is referenced by:  bj-df-ifc  32207
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