Theorem cbvralfw 2687

Description: Rule used to change bound variables, using implicit substitution. Version of cbvralf 2689 with a disjoint variable condition. Although we don't do so yet, we expect this disjoint variable condition will allow us to remove reliance on ax-i12 1500 and ax-bndl 1502 in the proof. (Contributed by NM, 7-Mar-2004.) (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 2306	. . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
3		cbvralfw.3	. . . 4 ⊢ Ⅎ𝑦𝜑
4	2, 3	nfim 1565	. . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 → 𝜑)
5		cbvralfw.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
6	5	nfcri 2306	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
7		cbvralfw.4	. . . 4 ⊢ Ⅎ𝑥𝜓
8	6, 7	nfim 1565	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓)
9		eleq1w 2231	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
10		cbvralfw.5	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
11	9, 10	imbi12d 233	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜓)))
12	4, 8, 11	cbvalv1 1744	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
13		df-ral 2453	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
14		df-ral 2453	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
15	12, 13, 14	3bitr4i 211	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1346  Ⅎwnf 1453   ∈ wcel 2141  Ⅎwnfc 2299  ∀wral 2448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453

This theorem is referenced by:  cbvralw  2691  nnwofdc  11993