Theorem cbvrab 3193

Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
cbvrab	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}

Proof of Theorem cbvrab

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1841	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝜑)
2		cbvrab.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2756	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
4		nfs1v 2435	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
5	3, 4	nfan 1826	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)
6		eleq1 2687	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
7		sbequ12 2109	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
8	6, 7	anbi12d 746	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)))
9	1, 5, 8	cbvab 2744	. . 3 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)}
10		cbvrab.2	. . . . . 6 ⊢ Ⅎ𝑦𝐴
11	10	nfcri 2756	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴
12		cbvrab.3	. . . . . 6 ⊢ Ⅎ𝑦𝜑
13	12	nfsb 2438	. . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
14	11, 13	nfan 1826	. . . 4 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)
15		nfv 1841	. . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 ∧ 𝜓)
16		eleq1 2687	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
17		sbequ 2374	. . . . . 6 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
18		cbvrab.4	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
19		cbvrab.5	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
20	18, 19	sbie 2406	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
21	17, 20	syl6bb 276	. . . . 5 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓))
22	16, 21	anbi12d 746	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓)))
23	14, 15, 22	cbvab 2744	. . 3 ⊢ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)}
24	9, 23	eqtri 2642	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)}
25		df-rab 2918	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
26		df-rab 2918	. 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜓} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)}
27	24, 25, 26	3eqtr4i 2652	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1481  Ⅎwnf 1706  [wsb 1878   ∈ wcel 1988  {cab 2606  Ⅎwnfc 2749  {crab 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918

This theorem is referenced by:  cbvrabv  3194  elrabsf  3468  tfis  7039  cantnflem1  8571  scottexs  8735  scott0s  8736  elmptrab  21611  bnj1534  30897  scottexf  33947  scott0f  33948  eq0rabdioph  37159  rexrabdioph  37177  rexfrabdioph  37178  elnn0rabdioph  37186  dvdsrabdioph  37193  binomcxplemdvsum  38374  fnlimcnv  39699  fnlimabslt  39711  stoweidlem34  40014  stoweidlem59  40039  pimltmnf2  40674  pimgtpnf2  40680  pimltpnf2  40686  issmff  40706  smfpimltxrmpt  40730  smfpreimagtf  40739  smflim  40748  smfpimgtxr  40751  smfpimgtxrmpt  40755  smflim2  40775  smflimsup  40797  smfliminf  40800