Theorem reueqbii 37021

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 2852	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3		reueqbii.2	. . . 4 ⊢ (𝜓 ↔ 𝜒)
4	2, 3	anbi12i 640	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))
5	4	eubii 2610	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜒))
6		df-reu 3366	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
7		df-reu 3366	. 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 401   = wceq 1569   ∈ wcel 2144  ∃!weu 2593  ∃!wreu 3363

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-mo 2564  df-eu 2594  df-cleq 2752  df-clel 2835  df-reu 3366

This theorem is used by: (None)