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Theorem cbvrab 2572
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 1437 . . . 4 𝑧(𝑥𝐴𝜑)
2 cbvrab.1 . . . . . 6 𝑥𝐴
32nfcri 2188 . . . . 5 𝑥 𝑧𝐴
4 nfs1v 1831 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
53, 4nfan 1473 . . . 4 𝑥(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
6 eleq1 2116 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
7 sbequ12 1670 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
86, 7anbi12d 450 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝜑) ↔ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)))
91, 5, 8cbvab 2176 . . 3 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑧 ∣ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)}
10 cbvrab.2 . . . . . 6 𝑦𝐴
1110nfcri 2188 . . . . 5 𝑦 𝑧𝐴
12 cbvrab.3 . . . . . 6 𝑦𝜑
1312nfsb 1838 . . . . 5 𝑦[𝑧 / 𝑥]𝜑
1411, 13nfan 1473 . . . 4 𝑦(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
15 nfv 1437 . . . 4 𝑧(𝑦𝐴𝜓)
16 eleq1 2116 . . . . 5 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
17 sbequ 1737 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
18 cbvrab.4 . . . . . . 7 𝑥𝜓
19 cbvrab.5 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
2018, 19sbie 1690 . . . . . 6 ([𝑦 / 𝑥]𝜑𝜓)
2117, 20syl6bb 189 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
2216, 21anbi12d 450 . . . 4 (𝑧 = 𝑦 → ((𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐴𝜓)))
2314, 15, 22cbvab 2176 . . 3 {𝑧 ∣ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)} = {𝑦 ∣ (𝑦𝐴𝜓)}
249, 23eqtri 2076 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦 ∣ (𝑦𝐴𝜓)}
25 df-rab 2332 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
26 df-rab 2332 . 2 {𝑦𝐴𝜓} = {𝑦 ∣ (𝑦𝐴𝜓)}
2724, 25, 263eqtr4i 2086 1 {𝑥𝐴𝜑} = {𝑦𝐴𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wnf 1365  wcel 1409  [wsb 1661  {cab 2042  wnfc 2181  {crab 2327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rab 2332
This theorem is referenced by:  cbvrabv  2573  elrabsf  2823  tfis  4333
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