Theorem bj-ififc 37175

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 37173	. . 3 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)}
2	1	eleq2i 2855	. 2 ⊢ (𝑋 ∈ if(𝜑, 𝐴, 𝐵) ↔ 𝑋 ∈ {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)})
3		df-ifp 1079	. . . 4 ⊢ (if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵) ↔ ((𝜑 ∧ 𝑋 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑋 ∈ 𝐵)))
4		elex 3476	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ V)
5	4	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ V)
6		elex 3476	. . . . . 6 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ V)
7	6	adantl 486	. . . . 5 ⊢ ((¬ 𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ V)
8	5, 7	jaoi 870	. . . 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 2851	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
11		eleq1 2851	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵))
12	10, 11	ifpbi23d 1096	. . 3 ⊢ (𝑥 = 𝑋 → (if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵) ↔ if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵)))
13	9, 12	elab3 3645	. 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 400   ∨ wo 860  if-wif 1078   = wceq 1570   ∈ wcel 2143  {cab 2741  Vcvv 3455  ifcif 4487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1079  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-if 4488

This theorem is referenced by: (None)