Theorem opelopaba

Description: The law of concretion. Theorem 9.5 of Quine p. 61. (Contributed by Mario Carneiro, 19-Dec-2013)

	Ref	Expression
Hypotheses	opelopaba.1	⊢ 𝐴 ∈ V
	opelopaba.2	⊢ 𝐵 ∈ V
	opelopaba.3	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
Assertion	opelopaba	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ 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	3	opelopabga	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ 𝜓 ) )
5	1 2 4	mp2an	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ 𝜓 )