Theorem brabga

Description: The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013)

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

Proof

Step	Hyp	Ref	Expression
1		opelopabga.1	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
2		brabga.2	⊢ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 }
3		df-br	⊢ ( 𝐴 𝑅 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑅 )
4	2	eleq2i	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑅 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } )
5	3 4	bitri	⊢ ( 𝐴 𝑅 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } )
6	1	opelopabga	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ 𝜓 ) )
7	5 6	syl5bb	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 𝑅 𝐵 ↔ 𝜓 ) )