Theorem bj-ififc 34028

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, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵))

Proof of Theorem bj-ififc

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-df-ifc 34026	. . 3 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)}
2	1	eleq2i 2881	. 2 ⊢ (𝑋 ∈ if(𝜑, 𝐴, 𝐵) ↔ 𝑋 ∈ {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)})
3		df-ifp 1059	. . . 4 ⊢ (if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵) ↔ ((𝜑 ∧ 𝑋 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑋 ∈ 𝐵)))
4		elex 3459	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ V)
5	4	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ V)
6		elex 3459	. . . . . 6 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ V)
7	6	adantl 485	. . . . 5 ⊢ ((¬ 𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ V)
8	5, 7	jaoi 854	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑋 ∈ 𝐵)) → 𝑋 ∈ V)
9	3, 8	sylbi 220	. . 3 ⊢ (if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵) → 𝑋 ∈ V)
10		eleq1 2877	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
11		eleq1 2877	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵))
12	10, 11	ifpbi23d 1077	. . 3 ⊢ (𝑥 = 𝑋 → (if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵) ↔ if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵)))
13	9, 12	elab3 3622	. 2 ⊢ (𝑋 ∈ {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)} ↔ if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵))
14	2, 13	bitri 278	1 ⊢ (𝑋 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   ∨ wo 844  if-wif 1058   = wceq 1538   ∈ wcel 2111  {cab 2776  Vcvv 3441  ifcif 4425

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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

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 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-if 4426

This theorem is referenced by: (None)