Theorem 2reu2rex

Description: Double restricted existential uniqueness, analogous to 2eu2ex . (Contributed by Alexander van der Vekens, 25-Jun-2017)

	Ref	Expression
Assertion	2reu2rex	⊢ ( ∃! 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		reurex	⊢ ( ∃! 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 )
2		reurex	⊢ ( ∃! 𝑦 ∈ 𝐵 𝜑 → ∃ 𝑦 ∈ 𝐵 𝜑 )
3	2	reximi	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 )
4	1 3	syl	⊢ ( ∃! 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 )