Theorem cbvrmodavw2 37072

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

Hypotheses

Ref	Expression
cbvrmodavw2.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
cbvrmodavw2.2	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbvrmodavw2	⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑦 ∈ 𝐵 𝜒))

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

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

Proof of Theorem cbvrmodavw2

Step	Hyp	Ref	Expression
1		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
2		cbvrmodavw2.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)
3	1, 2	eleq12d 2855	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
4		cbvrmodavw2.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
5	3, 4	anbi12d 644	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑦 ∈ 𝐵 ∧ 𝜒)))
6	5	cbvmodavw 37039	. 2 ⊢ (𝜑 → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃*𝑦(𝑦 ∈ 𝐵 ∧ 𝜒)))
7		df-rmo 3366	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
8		df-rmo 3366	. 2 ⊢ (∃*𝑦 ∈ 𝐵 𝜒 ↔ ∃*𝑦(𝑦 ∈ 𝐵 ∧ 𝜒))
9	6, 7, 8	3bitr4g 317	1 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑦 ∈ 𝐵 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃*wmo 2563  ∃*wrmo 3365

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 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2565  df-cleq 2753  df-clel 2836  df-rmo 3366

This theorem is used by:  cbvdisjdavw2  37078