Theorem cbvralfw 3278

Description: Rule used to change bound variables, using implicit substitution. Version of cbvralf 3332 with a disjoint variable condition, which does not require ax-10 2147, ax-13 2377. For a version not dependent on ax-11 2163 and ax-12, see cbvralvw 3216. (Contributed by NM, 7-Mar-2004.) Avoid ax-10 2147, ax-13 2377. (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 2891	. . . 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 2891	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
7		cbvralfw.4	. . . 4 ⊢ Ⅎ𝑥𝜓
8	6, 7	nfim 1898	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓)
9		eleq1w 2820	. . . 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 2346	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
13		df-ral 3053	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
14		df-ral 3053	. 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 2884  ∀wral 3052

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 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-nf 1786  df-clel 2812  df-nfc 2886  df-ral 3053

This theorem is referenced by:  cbvrexfw  3279  cbvralw  3280  reusv2lem4  5350  reusv2  5352  ffnfvf  7076  nnwof  12841  nnindf  32917  scottexf  38448  scott0f  38449  rsp3  38646  evth2f  45404  evthf  45416  fmptff  45656  supxrleubrnmptf  45838  stoweidlem14  46401  stoweidlem28  46415  stoweidlem59  46446