Theorem brab

Description: The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999)

	Ref	Expression
Hypotheses	opelopab.1	⊢ 𝐴 ∈ V
	opelopab.2	⊢ 𝐵 ∈ V
	opelopab.3	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	opelopab.4	⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
	brab.5	⊢ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 }
Assertion	brab	⊢ ( 𝐴 𝑅 𝐵 ↔ 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		opelopab.1	⊢ 𝐴 ∈ V
2		opelopab.2	⊢ 𝐵 ∈ V
3		opelopab.3	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
4		opelopab.4	⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
5		brab.5	⊢ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 }
6	3 4 5	brabg	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 𝑅 𝐵 ↔ 𝜒 ) )
7	1 2 6	mp2an	⊢ ( 𝐴 𝑅 𝐵 ↔ 𝜒 )