Theorem reueqbidv

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

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

Proof

Step	Hyp	Ref	Expression
1		reueqbidv.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		reueqbidv.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
3	1	eleq2d	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
4	3 2	anbi12d	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜒 ) ) )
5	4	eubidv	⊢ ( 𝜑 → ( ∃! 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	3bitr4g	⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐴 𝜓 ↔ ∃! 𝑥 ∈ 𝐵 𝜒 ) )