Theorem bj-brab2a1 37589

Description: "Unbounded" version of brab2a 5733. (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 3452	. . . . . 6 ⊢ 𝑥 ∈ V
4		vex 3452	. . . . . 6 ⊢ 𝑦 ∈ V
5	3, 4	pm3.2i 473	. . . . 5 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
6	5	biantrur 537	. . . 4 ⊢ (𝜑 ↔ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝜑))
7	6	opabbii 5161	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝜑)}
8	2, 7	eqtri 2779	. 2 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝜑)}
9	1, 8	brab2a 5733	1 ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1554   ∈ wcel 2136  Vcvv 3448   class class class wbr 5094  {copab 5156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-sep 5240  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-br 5095  df-opab 5157  df-xp 5646

This theorem is referenced by:  bj-ideqg1  37604