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Metamath Proof Explorer

Theorem eqbrrdv

Description: Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017)

Ref Expression
Hypotheses eqbrrdv.1 ⊢ ( 𝜑 → Rel 𝐴 )
eqbrrdv.2 ⊢ ( 𝜑 → Rel 𝐵 )
eqbrrdv.3 ⊢ ( 𝜑 → ( 𝑥 𝐴 𝑦 ↔ 𝑥 𝐵 𝑦 ) )
Assertion eqbrrdv ( 𝜑 → 𝐴 = 𝐵 )

Proof

Step Hyp Ref Expression
1 eqbrrdv.1 ⊢ ( 𝜑 → Rel 𝐴 )
2 eqbrrdv.2 ⊢ ( 𝜑 → Rel 𝐵 )
3 eqbrrdv.3 ⊢ ( 𝜑 → ( 𝑥 𝐴 𝑦 ↔ 𝑥 𝐵 𝑦 ) )
4 df-br ⊢ ( 𝑥 𝐴 𝑦 ↔ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 )
5 df-br ⊢ ( 𝑥 𝐵 𝑦 ↔ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 )
6 3 4 5 3bitr3g ⊢ ( 𝜑 → ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ↔ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) )
7 6 alrimivv ⊢ ( 𝜑 → ∀ 𝑥 ∀ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ↔ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) )
8 eqrel ⊢ ( ( Rel 𝐴 ∧ Rel 𝐵 ) → ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ∀ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ↔ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) ) )
9 1 2 8 syl2anc ⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ∀ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ↔ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) ) )
10 7 9 mpbird ⊢ ( 𝜑 → 𝐴 = 𝐵 )