Description: A version of eqrelrdv . (Contributed by Rodolfo Medina, 10-Oct-2010)
Ref | Expression | ||
---|---|---|---|
Hypothesis | eqrelrdv2.1 | ⊢ ( 𝜑 → ( 〈 𝑥 , 𝑦 〉 ∈ 𝐴 ↔ 〈 𝑥 , 𝑦 〉 ∈ 𝐵 ) ) | |
Assertion | eqrelrdv2 | ⊢ ( ( ( Rel 𝐴 ∧ Rel 𝐵 ) ∧ 𝜑 ) → 𝐴 = 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqrelrdv2.1 | ⊢ ( 𝜑 → ( 〈 𝑥 , 𝑦 〉 ∈ 𝐴 ↔ 〈 𝑥 , 𝑦 〉 ∈ 𝐵 ) ) | |
2 | 1 | alrimivv | ⊢ ( 𝜑 → ∀ 𝑥 ∀ 𝑦 ( 〈 𝑥 , 𝑦 〉 ∈ 𝐴 ↔ 〈 𝑥 , 𝑦 〉 ∈ 𝐵 ) ) |
3 | eqrel | ⊢ ( ( Rel 𝐴 ∧ Rel 𝐵 ) → ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ∀ 𝑦 ( 〈 𝑥 , 𝑦 〉 ∈ 𝐴 ↔ 〈 𝑥 , 𝑦 〉 ∈ 𝐵 ) ) ) | |
4 | 2 3 | syl5ibr | ⊢ ( ( Rel 𝐴 ∧ Rel 𝐵 ) → ( 𝜑 → 𝐴 = 𝐵 ) ) |
5 | 4 | imp | ⊢ ( ( ( Rel 𝐴 ∧ Rel 𝐵 ) ∧ 𝜑 ) → 𝐴 = 𝐵 ) |