Theorem reusng

Description: Restricted existential uniqueness over a singleton. (Contributed by AV, 3-Apr-2023)

	Ref	Expression
Hypothesis	ralsng.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	reusng	⊢ ( 𝐴 ∈ 𝑉 → ( ∃! 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		ralsng.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		nfv	⊢ Ⅎ 𝑥 𝜓
3	2 1	reusngf	⊢ ( 𝐴 ∈ 𝑉 → ( ∃! 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 ) )