Theorem reueqbii 36431

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 2833	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3		reueqbii.2	. . . 4 ⊢ (𝜓 ↔ 𝜒)
4	2, 3	anbi12i 635	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))
5	4	eubii 2591	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜒))
6		df-reu 3347	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
7		df-reu 3347	. 2 ⊢ (∃!𝑥 ∈ 𝐵 𝜒 ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜒))
8	5, 6, 7	3bitr4i 305	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜒)

Syntax hints:   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∃!weu 2574  ∃!wreu 3344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-mo 2545  df-eu 2575  df-cleq 2733  df-clel 2816  df-reu 3347

This theorem is referenced by: (None)