Theorem cbvrmovw2 36628

Description: Change bound variable and domain in the restricted at-most-one quantifier, using implicit substitution. (Contributed by GG, 14-Aug-2025.)

Hypotheses

Ref	Expression
cbvrmovw2.1	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
cbvrmovw2.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvrmovw2	⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐵 𝜓)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥   𝑦,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem cbvrmovw2

Step	Hyp	Ref	Expression
1		eleq1w 2852	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2		cbvrmovw2.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
3	2	eleq2d 2855	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
4	1, 3	bitrd 282	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
5		cbvrmovw2.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	4, 5	anbi12d 643	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐵 ∧ 𝜓)))
7	6	cbvmovw 2636	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑦(𝑦 ∈ 𝐵 ∧ 𝜓))
8		df-rmo 3376	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
9		df-rmo 3376	. 2 ⊢ (∃*𝑦 ∈ 𝐵 𝜓 ↔ ∃*𝑦(𝑦 ∈ 𝐵 ∧ 𝜓))
10	7, 8, 9	3bitr4i 306	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃*wmo 2571  ∃*wrmo 3375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-mo 2573  df-cleq 2761  df-clel 2844  df-rmo 3376

This theorem is referenced by:  cbvdisjvw2  36635