Theorem reueqbii

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

	Ref	Expression
Hypotheses	reueqbii.1	⊢ 𝐴 = 𝐵
	reueqbii.2	⊢ ( 𝜓 ↔ 𝜒 )
Assertion	reueqbii	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ∈ 𝐵 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		reueqbii.1	⊢ 𝐴 = 𝐵
2		reueqbii.2	⊢ ( 𝜓 ↔ 𝜒 )
3	1	eleq2i	⊢ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 )
4	3 2	anbi12i	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜒 ) )
5	4	eubii	⊢ ( ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜒 ) )
6		df-reu	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
7		df-reu	⊢ ( ∃! 𝑥 ∈ 𝐵 𝜒 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜒 ) )
8	5 6 7	3bitr4i	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ∈ 𝐵 𝜒 )