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Theorem cbvrab 3438
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. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker cbvrabw 3437 when possible. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.)
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 1915 . . . 4 𝑧(𝑥𝐴𝜑)
2 cbvrab.1 . . . . . 6 𝑥𝐴
32nfcri 2943 . . . . 5 𝑥 𝑧𝐴
4 nfs1v 2157 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
53, 4nfan 1900 . . . 4 𝑥(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
6 eleq1w 2872 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
7 sbequ12 2250 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
86, 7anbi12d 633 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝜑) ↔ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)))
91, 5, 8cbvab 2869 . . 3 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑧 ∣ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)}
10 cbvrab.2 . . . . . 6 𝑦𝐴
1110nfcri 2943 . . . . 5 𝑦 𝑧𝐴
12 cbvrab.3 . . . . . 6 𝑦𝜑
1312nfsb 2542 . . . . 5 𝑦[𝑧 / 𝑥]𝜑
1411, 13nfan 1900 . . . 4 𝑦(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
15 nfv 1915 . . . 4 𝑧(𝑦𝐴𝜓)
16 eleq1w 2872 . . . . 5 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
17 sbequ 2088 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
18 cbvrab.4 . . . . . . 7 𝑥𝜓
19 cbvrab.5 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
2018, 19sbie 2521 . . . . . 6 ([𝑦 / 𝑥]𝜑𝜓)
2117, 20syl6bb 290 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
2216, 21anbi12d 633 . . . 4 (𝑧 = 𝑦 → ((𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐴𝜓)))
2314, 15, 22cbvab 2869 . . 3 {𝑧 ∣ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)} = {𝑦 ∣ (𝑦𝐴𝜓)}
249, 23eqtri 2821 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦 ∣ (𝑦𝐴𝜓)}
25 df-rab 3115 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
26 df-rab 3115 . 2 {𝑦𝐴𝜓} = {𝑦 ∣ (𝑦𝐴𝜓)}
2724, 25, 263eqtr4i 2831 1 {𝑥𝐴𝜑} = {𝑦𝐴𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wnf 1785  [wsb 2069  wcel 2111  {cab 2776  wnfc 2936  {crab 3110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115
This theorem is referenced by: (None)
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