Theorem reueqi

Description: Equality inference for restricted existential uniqueness quantifier. (Contributed by GG, 1-Sep-2025)

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

Proof

Step	Hyp	Ref	Expression
1		reueqi.1	⊢ 𝐴 = 𝐵
2	1	eleq2i	⊢ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 )
3	2	anbi1i	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) )
4	3	eubii	⊢ ( ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) )
5		df-reu	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
6		df-reu	⊢ ( ∃! 𝑥 ∈ 𝐵 𝜓 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) )
7	4 5 6	3bitr4i	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ∈ 𝐵 𝜓 )