Theorem bj-brab2a1 37771

Description: "Unbounded" version of brab2a 5756. (Contributed by BJ, 25-Dec-2023.)

Hypotheses

Ref	Expression
bj-brab2a1.1	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
bj-brab2a1.2	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Assertion

Ref	Expression
bj-brab2a1	⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜓))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem bj-brab2a1

Step	Hyp	Ref	Expression
1		bj-brab2a1.1	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
2		bj-brab2a1.2	. . 3 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
3		vex 3459	. . . . . 6 ⊢ 𝑥 ∈ V
4		vex 3459	. . . . . 6 ⊢ 𝑦 ∈ V
5	3, 4	pm3.2i 475	. . . . 5 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
6	5	biantrur 539	. . . 4 ⊢ (𝜑 ↔ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝜑))
7	6	opabbii 5179	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝜑)}
8	2, 7	eqtri 2786	. 2 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝜑)}
9	1, 8	brab2a 5756	1 ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455   class class class wbr 5110  {copab 5174

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  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669

This theorem is referenced by:  bj-ideqg1  37786