Theorem cbvralcsf 3880

Description: A more general version of cbvralf 3325 that doesn't require 𝐴 and 𝐵 to be distinct from 𝑥 or 𝑦. Changes bound variables using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by Andrew Salmon, 13-Jul-2011.) (New usage is discouraged.)

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 1921	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 → 𝜑)
2		nfcsb1v 3862	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴
3	2	nfcri 2894	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴
4		nfsbc1v 3750	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
5	3, 4	nfim 1903	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 → [𝑧 / 𝑥]𝜑)
6		id 22	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
7		csbeq1a 3852	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
8	6, 7	eleq12d 2834	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴))
9		sbceq1a 3741	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
10	8, 9	imbi12d 345	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 → [𝑧 / 𝑥]𝜑)))
11	1, 5, 10	cbvalv1 2349	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 → [𝑧 / 𝑥]𝜑))
12		nfcv 2902	. . . . . . 7 ⊢ Ⅎ𝑦𝑧
13		cbvralcsf.1	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
14	12, 13	nfcsb 3865	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑧 / 𝑥⦌𝐴
15	14	nfcri 2894	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴
16		cbvralcsf.3	. . . . . 6 ⊢ Ⅎ𝑦𝜑
17	12, 16	nfsbc 3755	. . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
18	15, 17	nfim 1903	. . . 4 ⊢ Ⅎ𝑦(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 → [𝑧 / 𝑥]𝜑)
19		nfv 1921	. . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐵 → 𝜓)
20		id 22	. . . . . 6 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦)
21		csbeq1 3841	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐴)
22		df-csb 3839	. . . . . . . 8 ⊢ ⦋𝑦 / 𝑥⦌𝐴 = {𝑣 ∣ [𝑦 / 𝑥]𝑣 ∈ 𝐴}
23		cbvralcsf.2	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝐵
24	23	nfcri 2894	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑣 ∈ 𝐵
25		cbvralcsf.5	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
26	25	eleq2d 2826	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑣 ∈ 𝐴 ↔ 𝑣 ∈ 𝐵))
27	24, 26	sbie 2510	. . . . . . . . . 10 ⊢ ([𝑦 / 𝑥]𝑣 ∈ 𝐴 ↔ 𝑣 ∈ 𝐵)
28		sbsbc 3734	. . . . . . . . . 10 ⊢ ([𝑦 / 𝑥]𝑣 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑣 ∈ 𝐴)
29	27, 28	bitr3i 278	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝐵 ↔ [𝑦 / 𝑥]𝑣 ∈ 𝐴)
30	29	eqabi 2875	. . . . . . . 8 ⊢ 𝐵 = {𝑣 ∣ [𝑦 / 𝑥]𝑣 ∈ 𝐴}
31	22, 30	eqtr4i 2766	. . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝐴 = 𝐵
32	21, 31	eqtrdi 2791	. . . . . 6 ⊢ (𝑧 = 𝑦 → ⦋𝑧 / 𝑥⦌𝐴 = 𝐵)
33	20, 32	eleq12d 2834	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ↔ 𝑦 ∈ 𝐵))
34		dfsbcq 3732	. . . . . 6 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
35		sbsbc 3734	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
36		cbvralcsf.4	. . . . . . . 8 ⊢ Ⅎ𝑥𝜓
37		cbvralcsf.6	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
38	36, 37	sbie 2510	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
39	35, 38	bitr3i 278	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
40	34, 39	bitrdi 288	. . . . 5 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓))
41	33, 40	imbi12d 345	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 → [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐵 → 𝜓)))
42	18, 19, 41	cbvalv1 2349	. . 3 ⊢ (∀𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝜓))
43	11, 42	bitri 276	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝜓))
44		df-ral 3055	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
45		df-ral 3055	. 2 ⊢ (∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝜓))
46	43, 44, 45	3bitr4i 304	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1545   = wceq 1547  Ⅎwnf 1790  [wsb 2073   ∈ wcel 2119  {cab 2718  Ⅎwnfc 2887  ∀wral 3054  [wsbc 3730  ⦋csb 3838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-13 2380  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ral 3055  df-sbc 3731  df-csb 3839

This theorem is referenced by:  cbvrexcsf  3881  cbvralv2  3884