Theorem reueqi 36145

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 2836	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3	2	anbi1i 623	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))
4	3	eubii 2588	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
5		df-reu 3389	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
6		df-reu 3389	. 2 ⊢ (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
7	4, 5, 6	3bitr4i 303	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜓)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∃!weu 2571  ∃!wreu 3386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-mo 2543  df-eu 2572  df-cleq 2732  df-clel 2819  df-reu 3389

This theorem is referenced by: (None)