Theorem cbvralfw 3437

Description: Rule used to change bound variables, using implicit substitution. Version of cbvralf 3439 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 7-Mar-2004.) (Revised by Gino Giotto, 10-Jan-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

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1915	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 → 𝜑)
2		cbvralfw.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2971	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
4		nfs1v 2160	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
5	3, 4	nfim 1897	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑)
6		eleq1w 2895	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
7		sbequ12 2253	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
8	6, 7	imbi12d 347	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑)))
9	1, 5, 8	cbvalv1 2361	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑧(𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑))
10		cbvralfw.2	. . . . . 6 ⊢ Ⅎ𝑦𝐴
11	10	nfcri 2971	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴
12		cbvralfw.3	. . . . . 6 ⊢ Ⅎ𝑦𝜑
13	12	nfsbv 2349	. . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
14	11, 13	nfim 1897	. . . 4 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑)
15		nfv 1915	. . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 → 𝜓)
16		eleq1w 2895	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
17		sbequ 2090	. . . . . 6 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
18		cbvralfw.4	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
19		cbvralfw.5	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
20	18, 19	sbiev 2330	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
21	17, 20	syl6bb 289	. . . . 5 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓))
22	16, 21	imbi12d 347	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜓)))
23	14, 15, 22	cbvalv1 2361	. . 3 ⊢ (∀𝑧(𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
24	9, 23	bitri 277	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
25		df-ral 3143	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
26		df-ral 3143	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
27	24, 25, 26	3bitr4i 305	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535  Ⅎwnf 1784  [wsb 2069   ∈ wcel 2114  Ⅎwnfc 2961  ∀wral 3138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-10 2145  ax-11 2161  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070  df-clel 2893  df-nfc 2963  df-ral 3143

This theorem is referenced by:  cbvrexfw  3438  cbvralw  3441  reusv2lem4  5302  reusv2  5304  ffnfvf  6883  nnwof  12315  nnindf  30535  scottexf  35461  scott0f  35462  evth2f  41321  evthf  41333  supxrleubrnmptf  41776  stoweidlem14  42348  stoweidlem28  42362  stoweidlem59  42393