Theorem reueubd

Description: Restricted existential uniqueness is equivalent to existential uniqueness if the unique element is in the restricting class. (Contributed by AV, 4-Jan-2021)

	Ref	Expression
Hypothesis	reueubd.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝑥 ∈ 𝑉 )
Assertion	reueubd	⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝑉 𝜓 ↔ ∃! 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		reueubd.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝑥 ∈ 𝑉 )
2		df-reu	⊢ ( ∃! 𝑥 ∈ 𝑉 𝜓 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝑉 ∧ 𝜓 ) )
3	1	ex	⊢ ( 𝜑 → ( 𝜓 → 𝑥 ∈ 𝑉 ) )
4	3	pm4.71rd	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝑥 ∈ 𝑉 ∧ 𝜓 ) ) )
5	4	eubidv	⊢ ( 𝜑 → ( ∃! 𝑥 𝜓 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝑉 ∧ 𝜓 ) ) )
6	2 5	bitr4id	⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝑉 𝜓 ↔ ∃! 𝑥 𝜓 ) )