Theorem cbvralfw 3282

Description: Rule used to change bound variables, using implicit substitution. Version of cbvralf 3337 with a disjoint variable condition, which does not require ax-10 2140, ax-13 2375. For a version not dependent on ax-11 2156 and ax-12, see cbvralvw 3218. (Contributed by NM, 7-Mar-2004.) Avoid ax-10 2140, 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 1895	. . 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 1895	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓)
9		eleq1w 2816	. . . 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 2341	. 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 1537  Ⅎwnf 1782   ∈ wcel 2107  Ⅎwnfc 2882  ∀wral 3050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-11 2156  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783  df-clel 2808  df-nfc 2884  df-ral 3051

This theorem is referenced by:  cbvrexfw  3283  cbvralw  3284  reusv2lem4  5368  reusv2  5370  ffnfvf  7106  nnwof  12922  nnindf  32731  scottexf  38113  scott0f  38114  evth2f  44966  evthf  44978  fmptff  45220  supxrleubrnmptf  45406  stoweidlem14  45973  stoweidlem28  45987  stoweidlem59  46018