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Metamath Proof Explorer

Theorem brrabga

Description: The law of concretion for operation class abstraction. (Contributed by Peter Mazsa, 24-Oct-2022)

Ref Expression
Hypotheses brrabga.1 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( 𝜑 ↔ 𝜓 ) )
brrabga.2 ⊢ 𝑅 = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 }
Assertion brrabga ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( ⟨ 𝐴 , 𝐵 ⟩ 𝑅 𝐶 ↔ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 brrabga.1 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( 𝜑 ↔ 𝜓 ) )
2 brrabga.2 ⊢ 𝑅 = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 }
3 df-br ⊢ ( ⟨ 𝐴 , 𝐵 ⟩ 𝑅 𝐶 ↔ ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ 𝑅 )
4 2 eleq2i ⊢ ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ 𝑅 ↔ ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } )
5 3 4 bitri ⊢ ( ⟨ 𝐴 , 𝐵 ⟩ 𝑅 𝐶 ↔ ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } )
6 1 eloprabga ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } ↔ 𝜓 ) )
7 5 6 bitrid ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( ⟨ 𝐴 , 𝐵 ⟩ 𝑅 𝐶 ↔ 𝜓 ) )