Theorem braba

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

	Ref	Expression
Hypotheses	opelopaba.1	⊢ 𝐴 ∈ V
	opelopaba.2	⊢ 𝐵 ∈ V
	opelopaba.3	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
	braba.4	⊢ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 }
Assertion	braba	⊢ ( 𝐴 𝑅 𝐵 ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		opelopaba.1	⊢ 𝐴 ∈ V
2		opelopaba.2	⊢ 𝐵 ∈ V
3		opelopaba.3	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
4		braba.4	⊢ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 }
5	3 4	brabga	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 𝑅 𝐵 ↔ 𝜓 ) )
6	1 2 5	mp2an	⊢ ( 𝐴 𝑅 𝐵 ↔ 𝜓 )