Theorem rmoeqbii 36361

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 2827	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3		rmoeqbii.2	. . . 4 ⊢ (𝜓 ↔ 𝜒)
4	2, 3	anbi12i 629	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))
5	4	mobii 2547	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜒))
6		df-rmo 3349	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
7		df-rmo 3349	. 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜒 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜒))
8	5, 6, 7	3bitr4i 303	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜒)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃*wmo 2536  ∃*wrmo 3348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-mo 2538  df-cleq 2727  df-clel 2810  df-rmo 3349

This theorem is referenced by: (None)