Theorem 2eu2ex

Description: Double existential uniqueness. (Contributed by NM, 3-Dec-2001)

	Ref	Expression
Assertion	2eu2ex	⊢ ( ∃! 𝑥 ∃! 𝑦 𝜑 → ∃ 𝑥 ∃ 𝑦 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		euex	⊢ ( ∃! 𝑥 ∃! 𝑦 𝜑 → ∃ 𝑥 ∃! 𝑦 𝜑 )
2		euex	⊢ ( ∃! 𝑦 𝜑 → ∃ 𝑦 𝜑 )
3	2	eximi	⊢ ( ∃ 𝑥 ∃! 𝑦 𝜑 → ∃ 𝑥 ∃ 𝑦 𝜑 )
4	1 3	syl	⊢ ( ∃! 𝑥 ∃! 𝑦 𝜑 → ∃ 𝑥 ∃ 𝑦 𝜑 )