Theorem reueqi 36729

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

Hypothesis

Ref	Expression
reueqi.1	⊢ 𝐴 = 𝐵

Assertion

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

Proof of Theorem reueqi

Step	Hyp	Ref	Expression
1		reueqi.1	. . . . 5 ⊢ 𝐴 = 𝐵
2	1	eleq2i 2854	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3	2	anbi1i 635	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))
4	3	eubii 2612	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
5		df-reu 3369	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
6		df-reu 3369	. 2 ⊢ (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
7	4, 5, 6	3bitr4i 306	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜓)

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)