Theorem cbvralfw 3275

Description: Rule used to change bound variables, using implicit substitution. Version of cbvralf 3329 with a disjoint variable condition, which does not require ax-10 2147, ax-13 2375. For a version not dependent on ax-11 2163 and ax-12, see cbvralvw 3213. (Contributed by NM, 7-Mar-2004.) Avoid ax-10 2147, ax-13 2375. (Revised by GG, 23-May-2024.)

Hypotheses

Ref	Expression
cbvralfw.1	⊢ Ⅎ𝑥𝐴
cbvralfw.2	⊢ Ⅎ𝑦𝐴
cbvralfw.3	⊢ Ⅎ𝑦𝜑
cbvralfw.4	⊢ Ⅎ𝑥𝜓
cbvralfw.5	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvralfw	⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbvralfw

Step	Hyp	Ref	Expression
1		cbvralfw.2	. . . . 5 ⊢ Ⅎ𝑦𝐴
2	1	nfcri 2889	. . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
3		cbvralfw.3	. . . 4 ⊢ Ⅎ𝑦𝜑
4	2, 3	nfim 1898	. . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 → 𝜑)
5		cbvralfw.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
6	5	nfcri 2889	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
7		cbvralfw.4	. . . 4 ⊢ Ⅎ𝑥𝜓
8	6, 7	nfim 1898	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓)
9		eleq1w 2818	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
10		cbvralfw.5	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
11	9, 10	imbi12d 344	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜓)))
12	4, 8, 11	cbvalv1 2344	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
13		df-ral 3051	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
14		df-ral 3051	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
15	12, 13, 14	3bitr4i 303	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540  Ⅎwnf 1785   ∈ wcel 2114  Ⅎwnfc 2882  ∀wral 3050

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-11 2163  ax-12 2183

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-nf 1786  df-clel 2810  df-nfc 2884  df-ral 3051

This theorem is referenced by:  cbvrexfw  3276  cbvralw  3277  reusv2lem4  5345  reusv2  5347  ffnfvf  7065  nnwof  12829  nnindf  32879  scottexf  38338  scott0f  38339  rsp3  38536  evth2f  45297  evthf  45309  fmptff  45550  supxrleubrnmptf  45732  stoweidlem14  46295  stoweidlem28  46309  stoweidlem59  46340