Theorem cbvralvw2 36184

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

Hypotheses

Ref	Expression
cbvralvw2.1	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
cbvralvw2.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvralvw2	⊢ (∀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 cbvralvw2

Step	Hyp	Ref	Expression
1		eleq1w 2827	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2		cbvralvw2.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
3	2	eleq2d 2830	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
4	1, 3	bitrd 279	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
5		cbvralvw2.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	4, 5	imbi12d 344	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐵 → 𝜓)))
7	6	cbvalvw 2035	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝜓))
8		df-ral 3068	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
9		df-ral 3068	. 2 ⊢ (∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝜓))
10	7, 8, 9	3bitr4i 303	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535   = wceq 1537   ∈ wcel 2108  ∀wral 3067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-cleq 2732  df-clel 2819  df-ral 3068

This theorem is referenced by:  cbviinvw2  36191  cbvixpvw2  36203