Theorem cbvralfw 3285

Description: Rule used to change bound variables, using implicit substitution. Version of cbvralf 3331 with a disjoint variable condition, which does not require ax-10 2137, ax-13 2370. For a version not dependent on ax-11 2154 and ax-12, see cbvralvw 3223. (Contributed by NM, 7-Mar-2004.) Avoid ax-10 2137, ax-13 2370. (Revised by Gino Giotto, 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 1899	. . 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 1899	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓)
9		eleq1w 2815	. . . 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 2337	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
13		df-ral 3061	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
14		df-ral 3061	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
15	12, 13, 14	3bitr4i 302	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1539  Ⅎwnf 1785   ∈ wcel 2106  Ⅎwnfc 2882  ∀wral 3060

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-nf 1786  df-clel 2809  df-nfc 2884  df-ral 3061

This theorem is referenced by:  cbvrexfw  3286  cbvralw  3287  reusv2lem4  5361  reusv2  5363  ffnfvf  7072  nnwof  12848  nnindf  31785  scottexf  36700  scott0f  36701  evth2f  43342  evthf  43354  fmptff  43619  supxrleubrnmptf  43806  stoweidlem14  44375  stoweidlem28  44389  stoweidlem59  44420