Theorem reueqbii 36730

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

Hypotheses

Ref	Expression
reueqbii.1	⊢ 𝐴 = 𝐵
reueqbii.2	⊢ (𝜓 ↔ 𝜒)

Assertion

Ref	Expression
reueqbii	⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜒)

Proof of Theorem reueqbii

Step	Hyp	Ref	Expression
1		reueqbii.1	. . . . 5 ⊢ 𝐴 = 𝐵
2	1	eleq2i 2854	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3		reueqbii.2	. . . 4 ⊢ (𝜓 ↔ 𝜒)
4	2, 3	anbi12i 639	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))
5	4	eubii 2612	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜒))
6		df-reu 3369	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
7		df-reu 3369	. 2 ⊢ (∃!𝑥 ∈ 𝐵 𝜒 ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜒))
8	5, 6, 7	3bitr4i 306	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜒)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∃!weu 2595  ∃!wreu 3366

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-mo 2566  df-eu 2596  df-cleq 2754  df-clel 2837  df-reu 3369

This theorem is used by: (None)