Theorem oprabbi

Description: Equality deduction for class abstraction of nested ordered pairs. (Contributed by Giovanni Mascellani, 10-Apr-2018)

	Ref	Expression
Assertion	oprabbi	⊢ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( 𝜑 ↔ 𝜓 ) → { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } )

Proof

Step	Hyp	Ref	Expression
1		eqoprab2b	⊢ ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( 𝜑 ↔ 𝜓 ) )
2	1	biimpri	⊢ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( 𝜑 ↔ 𝜓 ) → { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } )