Description: "Unbounded" version of brab2a . (Contributed by BJ, 25-Dec-2023)
Ref | Expression | ||
---|---|---|---|
Hypotheses | bj-brab2a1.1 | ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) ) | |
bj-brab2a1.2 | ⊢ 𝑅 = { 〈 𝑥 , 𝑦 〉 ∣ 𝜑 } | ||
Assertion | bj-brab2a1 | ⊢ ( 𝐴 𝑅 𝐵 ↔ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ∧ 𝜓 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-brab2a1.1 | ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) ) | |
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 ) ) |