Theorem rmo5

Description: Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017)

	Ref	Expression
Assertion	rmo5	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜑 ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃! 𝑥 ∈ 𝐴 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		moeu	⊢ ( ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
2		df-rmo	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜑 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
3		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
4		df-reu	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
5	3 4	imbi12i	⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃! 𝑥 ∈ 𝐴 𝜑 ) ↔ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
6	1 2 5	3bitr4i	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜑 ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃! 𝑥 ∈ 𝐴 𝜑 ) )