Theorem cbvrabw 3424

Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. Version of cbvrab 3425 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Gino Giotto, 10-Jan-2024.)

Hypotheses

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

Assertion

Ref	Expression
cbvrabw	⊢ {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 cbvrabw

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1917	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝜑)
2		cbvrabw.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2894	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
4		nfs1v 2153	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
5	3, 4	nfan 1902	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)
6		eleq1w 2821	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
7		sbequ12 2244	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
8	6, 7	anbi12d 631	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)))
9	1, 5, 8	cbvabw 2812	. . 3 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)}
10		cbvrabw.2	. . . . . 6 ⊢ Ⅎ𝑦𝐴
11	10	nfcri 2894	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴
12		cbvrabw.3	. . . . . 6 ⊢ Ⅎ𝑦𝜑
13	12	nfsbv 2324	. . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
14	11, 13	nfan 1902	. . . 4 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)
15		nfv 1917	. . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 ∧ 𝜓)
16		eleq1w 2821	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
17		sbequ 2086	. . . . . 6 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
18		cbvrabw.4	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
19		cbvrabw.5	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
20	18, 19	sbiev 2309	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
21	17, 20	bitrdi 287	. . . . 5 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓))
22	16, 21	anbi12d 631	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓)))
23	14, 15, 22	cbvabw 2812	. . 3 ⊢ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)}
24	9, 23	eqtri 2766	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)}
25		df-rab 3073	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
26		df-rab 3073	. 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜓} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)}
27	24, 25, 26	3eqtr4i 2776	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  Ⅎwnf 1786  [wsb 2067   ∈ wcel 2106  {cab 2715  Ⅎwnfc 2887  {crab 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rab 3073

This theorem is referenced by:  elrabsf  3764  f1ossf1o  7000  tfis  7701  cantnflem1  9447  scottexs  9645  scott0s  9646  elmptrab  22978  bnj1534  32833  scottexf  36326  scott0f  36327  eq0rabdioph  40598  rexrabdioph  40616  rexfrabdioph  40617  elnn0rabdioph  40625  dvdsrabdioph  40632  binomcxplemdvsum  41973  fnlimcnv  43208  fnlimabslt  43220  stoweidlem34  43575  stoweidlem59  43600  pimltmnf2f  44235  pimgtpnf2f  44242  pimltpnf2f  44249  issmff  44270  smfpimltxrmpt  44294  smfpreimagtf  44303  smflim  44312  smfpimgtxr  44315  smfpimgtxrmpt  44319  smflim2  44339  smflimsup  44361  smfliminf  44364