Theorem eqopab2bw

Description: Equivalence of ordered pair abstraction equality and biconditional. Version of eqopab2b with a disjoint variable condition, which does not require ax-13 . (Contributed by Mario Carneiro, 4-Jan-2017) Avoid ax-13 . (Revised by GG, 26-Jan-2024)

	Ref	Expression
Assertion	eqopab2bw	⊢ ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ↔ ∀ 𝑥 ∀ 𝑦 ( 𝜑 ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		ssopab2bw	⊢ ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ⊆ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ↔ ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) )
2		ssopab2bw	⊢ ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ⊆ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ ∀ 𝑥 ∀ 𝑦 ( 𝜓 → 𝜑 ) )
3	1 2	anbi12i	⊢ ( ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ⊆ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ∧ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ⊆ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ) ↔ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ∧ ∀ 𝑥 ∀ 𝑦 ( 𝜓 → 𝜑 ) ) )
4		eqss	⊢ ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ↔ ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ⊆ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ∧ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ⊆ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ) )
5		2albiim	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ↔ 𝜓 ) ↔ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ∧ ∀ 𝑥 ∀ 𝑦 ( 𝜓 → 𝜑 ) ) )
6	3 4 5	3bitr4i	⊢ ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ↔ ∀ 𝑥 ∀ 𝑦 ( 𝜑 ↔ 𝜓 ) )