Theorem cbvralf 2830

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)

Hypotheses

Ref	Expression
cbvralf.1	⊢ ℲxA
cbvralf.2	⊢ ℲyA
cbvralf.3	⊢ Ⅎyφ
cbvralf.4	⊢ Ⅎxψ
cbvralf.5	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
cbvralf	⊢ (∀x ∈ A φ ↔ ∀y ∈ A ψ)

Proof of Theorem cbvralf

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . . 4 ⊢ Ⅎz(x ∈ A → φ)
2		cbvralf.1	. . . . . 6 ⊢ ℲxA
3	2	nfcri 2484	. . . . 5 ⊢ Ⅎx z ∈ A
4		nfs1v 2106	. . . . 5 ⊢ Ⅎx[z / x]φ
5	3, 4	nfim 1813	. . . 4 ⊢ Ⅎx(z ∈ A → [z / x]φ)
6		eleq1 2413	. . . . 5 ⊢ (x = z → (x ∈ A ↔ z ∈ A))
7		sbequ12 1919	. . . . 5 ⊢ (x = z → (φ ↔ [z / x]φ))
8	6, 7	imbi12d 311	. . . 4 ⊢ (x = z → ((x ∈ A → φ) ↔ (z ∈ A → [z / x]φ)))
9	1, 5, 8	cbval 1984	. . 3 ⊢ (∀x(x ∈ A → φ) ↔ ∀z(z ∈ A → [z / x]φ))
10		cbvralf.2	. . . . . 6 ⊢ ℲyA
11	10	nfcri 2484	. . . . 5 ⊢ Ⅎy z ∈ A
12		cbvralf.3	. . . . . 6 ⊢ Ⅎyφ
13	12	nfsb 2109	. . . . 5 ⊢ Ⅎy[z / x]φ
14	11, 13	nfim 1813	. . . 4 ⊢ Ⅎy(z ∈ A → [z / x]φ)
15		nfv 1619	. . . 4 ⊢ Ⅎz(y ∈ A → ψ)
16		eleq1 2413	. . . . 5 ⊢ (z = y → (z ∈ A ↔ y ∈ A))
17		sbequ 2060	. . . . . 6 ⊢ (z = y → ([z / x]φ ↔ [y / x]φ))
18		cbvralf.4	. . . . . . 7 ⊢ Ⅎxψ
19		cbvralf.5	. . . . . . 7 ⊢ (x = y → (φ ↔ ψ))
20	18, 19	sbie 2038	. . . . . 6 ⊢ ([y / x]φ ↔ ψ)
21	17, 20	syl6bb 252	. . . . 5 ⊢ (z = y → ([z / x]φ ↔ ψ))
22	16, 21	imbi12d 311	. . . 4 ⊢ (z = y → ((z ∈ A → [z / x]φ) ↔ (y ∈ A → ψ)))
23	14, 15, 22	cbval 1984	. . 3 ⊢ (∀z(z ∈ A → [z / x]φ) ↔ ∀y(y ∈ A → ψ))
24	9, 23	bitri 240	. 2 ⊢ (∀x(x ∈ A → φ) ↔ ∀y(y ∈ A → ψ))
25		df-ral 2620	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
26		df-ral 2620	. 2 ⊢ (∀y ∈ A ψ ↔ ∀y(y ∈ A → ψ))
27	24, 25, 26	3bitr4i 268	1 ⊢ (∀x ∈ A φ ↔ ∀y ∈ A ψ)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642  [wsb 1648   ∈ wcel 1710  Ⅎwnfc 2477  ∀wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620

This theorem is referenced by:  cbvrexf  2831  cbvral  2832  ffnfvf  5429