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Theorem bj-ififc 33989
 Description: A biconditional connecting the conditional operator for propositions and the conditional operator for classes. Note that there is no sethood hypothesis on 𝑋: it is implied by either side. (Contributed by BJ, 24-Sep-2019.) Generalize statement from setvar 𝑥 to class 𝑋. (Revised by BJ, 26-Dec-2023.)
Assertion
Ref Expression
bj-ififc (𝑋 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑋𝐴, 𝑋𝐵))

Proof of Theorem bj-ififc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 bj-df-ifc 33987 . . 3 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)}
21eleq2i 2905 . 2 (𝑋 ∈ if(𝜑, 𝐴, 𝐵) ↔ 𝑋 ∈ {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)})
3 df-ifp 1059 . . . 4 (if-(𝜑, 𝑋𝐴, 𝑋𝐵) ↔ ((𝜑𝑋𝐴) ∨ (¬ 𝜑𝑋𝐵)))
4 elex 3487 . . . . . 6 (𝑋𝐴𝑋 ∈ V)
54adantl 485 . . . . 5 ((𝜑𝑋𝐴) → 𝑋 ∈ V)
6 elex 3487 . . . . . 6 (𝑋𝐵𝑋 ∈ V)
76adantl 485 . . . . 5 ((¬ 𝜑𝑋𝐵) → 𝑋 ∈ V)
85, 7jaoi 854 . . . 4 (((𝜑𝑋𝐴) ∨ (¬ 𝜑𝑋𝐵)) → 𝑋 ∈ V)
93, 8sylbi 220 . . 3 (if-(𝜑, 𝑋𝐴, 𝑋𝐵) → 𝑋 ∈ V)
10 eleq1 2901 . . . 4 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
11 eleq1 2901 . . . 4 (𝑥 = 𝑋 → (𝑥𝐵𝑋𝐵))
1210, 11ifpbi23d 1077 . . 3 (𝑥 = 𝑋 → (if-(𝜑, 𝑥𝐴, 𝑥𝐵) ↔ if-(𝜑, 𝑋𝐴, 𝑋𝐵)))
139, 12elab3 3649 . 2 (𝑋 ∈ {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)} ↔ if-(𝜑, 𝑋𝐴, 𝑋𝐵))
142, 13bitri 278 1 (𝑋 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑋𝐴, 𝑋𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   ∨ wo 844  if-wif 1058   = wceq 1538   ∈ wcel 2114  {cab 2800  Vcvv 3469  ifcif 4439 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-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-v 3471  df-if 4440 This theorem is referenced by: (None)
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