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Theorem bj-ififc 34749
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 34747 . . 3 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)}
21eleq2i 2830 . 2 (𝑋 ∈ if(𝜑, 𝐴, 𝐵) ↔ 𝑋 ∈ {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)})
3 df-ifp 1061 . . . 4 (if-(𝜑, 𝑋𝐴, 𝑋𝐵) ↔ ((𝜑𝑋𝐴) ∨ (¬ 𝜑𝑋𝐵)))
4 elex 3448 . . . . . 6 (𝑋𝐴𝑋 ∈ V)
54adantl 482 . . . . 5 ((𝜑𝑋𝐴) → 𝑋 ∈ V)
6 elex 3448 . . . . . 6 (𝑋𝐵𝑋 ∈ V)
76adantl 482 . . . . 5 ((¬ 𝜑𝑋𝐵) → 𝑋 ∈ V)
85, 7jaoi 854 . . . 4 (((𝜑𝑋𝐴) ∨ (¬ 𝜑𝑋𝐵)) → 𝑋 ∈ V)
93, 8sylbi 216 . . 3 (if-(𝜑, 𝑋𝐴, 𝑋𝐵) → 𝑋 ∈ V)
10 eleq1 2826 . . . 4 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
11 eleq1 2826 . . . 4 (𝑥 = 𝑋 → (𝑥𝐵𝑋𝐵))
1210, 11ifpbi23d 1079 . . 3 (𝑥 = 𝑋 → (if-(𝜑, 𝑥𝐴, 𝑥𝐵) ↔ if-(𝜑, 𝑋𝐴, 𝑋𝐵)))
139, 12elab3 3617 . 2 (𝑋 ∈ {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)} ↔ if-(𝜑, 𝑋𝐴, 𝑋𝐵))
142, 13bitri 274 1 (𝑋 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑋𝐴, 𝑋𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396  wo 844  if-wif 1060   = wceq 1539  wcel 2106  {cab 2715  Vcvv 3430  ifcif 4460
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ifp 1061  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3432  df-if 4461
This theorem is referenced by: (None)
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