Theorem 2reu2reu2

Description: Double restricted existential uniqueness implies two nested restricted existential uniqueness. (Contributed by AV, 5-Jul-2023)

	Ref	Expression
Assertion	2reu2reu2	⊢ ( ∃! 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		2rexreu	⊢ ( ( ∃! 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ∧ ∃! 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ 𝐴 𝜑 ) → ∃! 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 )
3	1 2	sylbi	⊢ ( ∃! 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 𝜑 → ∃! 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 )