Theorem 2reu2rex1

Description: Double restricted existential uniqueness implies double restricted existence. (Contributed by Thierry Arnoux, 4-Jul-2023)

	Ref	Expression
Assertion	2reu2rex1	⊢ ( ∃! 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		df-2reu	⊢ ( ∃! 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 𝜑 ↔ ( ∃! 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ∧ ∃! 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ 𝐴 𝜑 ) )
2	1	simplbi	⊢ ( ∃! 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 𝜑 → ∃! 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 )
3		reurex	⊢ ( ∃! 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 )
4	2 3	syl	⊢ ( ∃! 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 )