Theorem reurmo

Description: Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017)

	Ref	Expression
Assertion	reurmo	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 → ∃* 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		reu5	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃* 𝑥 ∈ 𝐴 𝜑 ) )
2	1	simprbi	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 → ∃* 𝑥 ∈ 𝐴 𝜑 )