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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 𝑥𝐴
cbvrab.2 𝑦𝐴
cbvrab.3 𝑦𝜑
cbvrab.4 𝑥𝜓
cbvrab.5 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrab {𝑥𝐴𝜑} = {𝑦𝐴𝜓}

Proof of Theorem cbvrab
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1841 . . . 4 𝑧(𝑥𝐴𝜑)
2 cbvrab.1 . . . . . 6 𝑥𝐴
32nfcri 2756 . . . . 5 𝑥 𝑧𝐴
4 nfs1v 2435 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
53, 4nfan 1826 . . . 4 𝑥(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
6 eleq1 2687 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
7 sbequ12 2109 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
86, 7anbi12d 746 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝜑) ↔ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)))
91, 5, 8cbvab 2744 . . 3 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑧 ∣ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)}
10 cbvrab.2 . . . . . 6 𝑦𝐴
1110nfcri 2756 . . . . 5 𝑦 𝑧𝐴
12 cbvrab.3 . . . . . 6 𝑦𝜑
1312nfsb 2438 . . . . 5 𝑦[𝑧 / 𝑥]𝜑
1411, 13nfan 1826 . . . 4 𝑦(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
15 nfv 1841 . . . 4 𝑧(𝑦𝐴𝜓)
16 eleq1 2687 . . . . 5 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
17 sbequ 2374 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
18 cbvrab.4 . . . . . . 7 𝑥𝜓
19 cbvrab.5 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
2018, 19sbie 2406 . . . . . 6 ([𝑦 / 𝑥]𝜑𝜓)
2117, 20syl6bb 276 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
2216, 21anbi12d 746 . . . 4 (𝑧 = 𝑦 → ((𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐴𝜓)))
2314, 15, 22cbvab 2744 . . 3 {𝑧 ∣ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)} = {𝑦 ∣ (𝑦𝐴𝜓)}
249, 23eqtri 2642 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦 ∣ (𝑦𝐴𝜓)}
25 df-rab 2918 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
26 df-rab 2918 . 2 {𝑦𝐴𝜓} = {𝑦 ∣ (𝑦𝐴𝜓)}
2724, 25, 263eqtr4i 2652 1 {𝑥𝐴𝜑} = {𝑦𝐴𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wnf 1706  [wsb 1878  wcel 1988  {cab 2606  wnfc 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
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