Theorem opelopab

Description: The law of concretion. Theorem 9.5 of Quine p. 61. (Contributed by NM, 16-May-1995)

	Ref	Expression
Hypotheses	opelopab.1	⊢ 𝐴 ∈ V
	opelopab.2	⊢ 𝐵 ∈ V
	opelopab.3	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	opelopab.4	⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
Assertion	opelopab	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ 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	3 4	opelopabg	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ 𝜒 ) )
6	1 2 5	mp2an	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ 𝜒 )