Theorem rmoeqbii 36757

Description: Equality inference for restricted at-most-one quantifier. (Contributed by GG, 1-Sep-2025.)

Hypotheses

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

Assertion

Ref	Expression
rmoeqbii	⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜒)

Proof of Theorem rmoeqbii

Step	Hyp	Ref	Expression
1		rmoeqbii.1	. . . . 5 ⊢ 𝐴 = 𝐵
2	1	eleq2i 2857	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3		rmoeqbii.2	. . . 4 ⊢ (𝜓 ↔ 𝜒)
4	2, 3	anbi12i 640	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))
5	4	mobii 2578	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜒))
6		df-rmo 3371	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
7		df-rmo 3371	. 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 1570   ∈ wcel 2146  ∃*wmo 2567  ∃*wrmo 3370

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2569  df-cleq 2757  df-clel 2840  df-rmo 3371

This theorem is used by: (None)