Theorem cbvralcsf 3199

Description: A more general version of cbvralf 2830 that doesn't require A and B to be distinct from x or y. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)

Hypotheses

Ref	Expression
cbvralcsf.1	⊢ ℲyA
cbvralcsf.2	⊢ ℲxB
cbvralcsf.3	⊢ Ⅎyφ
cbvralcsf.4	⊢ Ⅎxψ
cbvralcsf.5	⊢ (x = y → A = B)
cbvralcsf.6	⊢ (x = y → (φ ↔ ψ))

Assertion

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

Proof of Theorem cbvralcsf

Dummy variables v z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . . 4 ⊢ Ⅎz(x ∈ A → φ)
2		nfcsb1v 3169	. . . . . 6 ⊢ Ⅎx[z / x]A
3	2	nfcri 2484	. . . . 5 ⊢ Ⅎx z ∈ [z / x]A
4		nfsbc1v 3066	. . . . 5 ⊢ Ⅎx[̣z / x]̣φ
5	3, 4	nfim 1813	. . . 4 ⊢ Ⅎx(z ∈ [z / x]A → [̣z / x]̣φ)
6		id 19	. . . . . 6 ⊢ (x = z → x = z)
7		csbeq1a 3145	. . . . . 6 ⊢ (x = z → A = [z / x]A)
8	6, 7	eleq12d 2421	. . . . 5 ⊢ (x = z → (x ∈ A ↔ z ∈ [z / x]A))
9		sbceq1a 3057	. . . . 5 ⊢ (x = z → (φ ↔ [̣z / x]̣φ))
10	8, 9	imbi12d 311	. . . 4 ⊢ (x = z → ((x ∈ A → φ) ↔ (z ∈ [z / x]A → [̣z / x]̣φ)))
11	1, 5, 10	cbval 1984	. . 3 ⊢ (∀x(x ∈ A → φ) ↔ ∀z(z ∈ [z / x]A → [̣z / x]̣φ))
12		nfcv 2490	. . . . . . 7 ⊢ Ⅎyz
13		cbvralcsf.1	. . . . . . 7 ⊢ ℲyA
14	12, 13	nfcsb 3171	. . . . . 6 ⊢ Ⅎy[z / x]A
15	14	nfcri 2484	. . . . 5 ⊢ Ⅎy z ∈ [z / x]A
16		cbvralcsf.3	. . . . . 6 ⊢ Ⅎyφ
17	12, 16	nfsbc 3068	. . . . 5 ⊢ Ⅎy[̣z / x]̣φ
18	15, 17	nfim 1813	. . . 4 ⊢ Ⅎy(z ∈ [z / x]A → [̣z / x]̣φ)
19		nfv 1619	. . . 4 ⊢ Ⅎz(y ∈ B → ψ)
20		id 19	. . . . . 6 ⊢ (z = y → z = y)
21		csbeq1 3140	. . . . . . 7 ⊢ (z = y → [z / x]A = [y / x]A)
22		df-csb 3138	. . . . . . . 8 ⊢ [y / x]A = {v ∣ [̣y / x]̣v ∈ A}
23		cbvralcsf.2	. . . . . . . . . . . 12 ⊢ ℲxB
24	23	nfcri 2484	. . . . . . . . . . 11 ⊢ Ⅎx v ∈ B
25		cbvralcsf.5	. . . . . . . . . . . 12 ⊢ (x = y → A = B)
26	25	eleq2d 2420	. . . . . . . . . . 11 ⊢ (x = y → (v ∈ A ↔ v ∈ B))
27	24, 26	sbie 2038	. . . . . . . . . 10 ⊢ ([y / x]v ∈ A ↔ v ∈ B)
28		sbsbc 3051	. . . . . . . . . 10 ⊢ ([y / x]v ∈ A ↔ [̣y / x]̣v ∈ A)
29	27, 28	bitr3i 242	. . . . . . . . 9 ⊢ (v ∈ B ↔ [̣y / x]̣v ∈ A)
30	29	eqabi 2465	. . . . . . . 8 ⊢ B = {v ∣ [̣y / x]̣v ∈ A}
31	22, 30	eqtr4i 2376	. . . . . . 7 ⊢ [y / x]A = B
32	21, 31	syl6eq 2401	. . . . . 6 ⊢ (z = y → [z / x]A = B)
33	20, 32	eleq12d 2421	. . . . 5 ⊢ (z = y → (z ∈ [z / x]A ↔ y ∈ B))
34		dfsbcq 3049	. . . . . 6 ⊢ (z = y → ([̣z / x]̣φ ↔ [̣y / x]̣φ))
35		sbsbc 3051	. . . . . . 7 ⊢ ([y / x]φ ↔ [̣y / x]̣φ)
36		cbvralcsf.4	. . . . . . . 8 ⊢ Ⅎxψ
37		cbvralcsf.6	. . . . . . . 8 ⊢ (x = y → (φ ↔ ψ))
38	36, 37	sbie 2038	. . . . . . 7 ⊢ ([y / x]φ ↔ ψ)
39	35, 38	bitr3i 242	. . . . . 6 ⊢ ([̣y / x]̣φ ↔ ψ)
40	34, 39	syl6bb 252	. . . . 5 ⊢ (z = y → ([̣z / x]̣φ ↔ ψ))
41	33, 40	imbi12d 311	. . . 4 ⊢ (z = y → ((z ∈ [z / x]A → [̣z / x]̣φ) ↔ (y ∈ B → ψ)))
42	18, 19, 41	cbval 1984	. . 3 ⊢ (∀z(z ∈ [z / x]A → [̣z / x]̣φ) ↔ ∀y(y ∈ B → ψ))
43	11, 42	bitri 240	. 2 ⊢ (∀x(x ∈ A → φ) ↔ ∀y(y ∈ B → ψ))
44		df-ral 2620	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
45		df-ral 2620	. 2 ⊢ (∀y ∈ B ψ ↔ ∀y(y ∈ B → ψ))
46	43, 44, 45	3bitr4i 268	1 ⊢ (∀x ∈ A φ ↔ ∀y ∈ B ψ)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477  ∀wral 2615  [̣wsbc 3047  [csb 3137

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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-sbc 3048  df-csb 3138

This theorem is referenced by:  cbvrexcsf  3200  cbvralv2  3203