Metamath Proof Explorer


Theorem bj-brab2a1

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

Ref Expression
Hypotheses bj-brab2a1.1 ( ( 𝑥 = 𝐴𝑦 = 𝐵 ) → ( 𝜑𝜓 ) )
bj-brab2a1.2 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 }
Assertion bj-brab2a1 ( 𝐴 𝑅 𝐵 ↔ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ∧ 𝜓 ) )

Proof

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