Theorem reueqdv

Description: Formula-building rule for restricted existential uniqueness quantifier. Deduction form. (Contributed by GG, 1-Sep-2025)

	Ref	Expression
Hypothesis	reueqdv.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	reueqdv	⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ∈ 𝐵 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		reueqdv.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		reueq1	⊢ ( 𝐴 = 𝐵 → ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ∈ 𝐵 𝜓 ) )
3	1 2	syl	⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ∈ 𝐵 𝜓 ) )