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Theorem cbvrabcsfw 3869
 Description: Version of cbvrabcsf 3873 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by Andrew Salmon, 13-Jul-2011.) (Revised by Gino Giotto, 26-Jan-2024.)
Hypotheses
Ref Expression
cbvrabcsfw.1 𝑦𝐴
cbvrabcsfw.2 𝑥𝐵
cbvrabcsfw.3 𝑦𝜑
cbvrabcsfw.4 𝑥𝜓
cbvrabcsfw.5 (𝑥 = 𝑦𝐴 = 𝐵)
cbvrabcsfw.6 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrabcsfw {𝑥𝐴𝜑} = {𝑦𝐵𝜓}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem cbvrabcsfw
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1915 . . . 4 𝑧(𝑥𝐴𝜑)
2 nfcsb1v 3852 . . . . . 6 𝑥𝑧 / 𝑥𝐴
32nfcri 2943 . . . . 5 𝑥 𝑧𝑧 / 𝑥𝐴
4 nfs1v 2157 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
53, 4nfan 1900 . . . 4 𝑥(𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑)
6 id 22 . . . . . 6 (𝑥 = 𝑧𝑥 = 𝑧)
7 csbeq1a 3842 . . . . . 6 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
86, 7eleq12d 2884 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝑧 / 𝑥𝐴))
9 sbequ12 2250 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
108, 9anbi12d 633 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝜑) ↔ (𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑)))
111, 5, 10cbvabw 2867 . . 3 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑧 ∣ (𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑)}
12 nfcv 2955 . . . . . . 7 𝑦𝑧
13 cbvrabcsfw.1 . . . . . . 7 𝑦𝐴
1412, 13nfcsbw 3854 . . . . . 6 𝑦𝑧 / 𝑥𝐴
1514nfcri 2943 . . . . 5 𝑦 𝑧𝑧 / 𝑥𝐴
16 cbvrabcsfw.3 . . . . . 6 𝑦𝜑
1716nfsbv 2338 . . . . 5 𝑦[𝑧 / 𝑥]𝜑
1815, 17nfan 1900 . . . 4 𝑦(𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑)
19 nfv 1915 . . . 4 𝑧(𝑦𝐵𝜓)
20 id 22 . . . . . 6 (𝑧 = 𝑦𝑧 = 𝑦)
21 csbeq1 3831 . . . . . . 7 (𝑧 = 𝑦𝑧 / 𝑥𝐴 = 𝑦 / 𝑥𝐴)
22 vex 3444 . . . . . . . 8 𝑦 ∈ V
23 cbvrabcsfw.2 . . . . . . . 8 𝑥𝐵
24 cbvrabcsfw.5 . . . . . . . 8 (𝑥 = 𝑦𝐴 = 𝐵)
2522, 23, 24csbief 3862 . . . . . . 7 𝑦 / 𝑥𝐴 = 𝐵
2621, 25eqtrdi 2849 . . . . . 6 (𝑧 = 𝑦𝑧 / 𝑥𝐴 = 𝐵)
2720, 26eleq12d 2884 . . . . 5 (𝑧 = 𝑦 → (𝑧𝑧 / 𝑥𝐴𝑦𝐵))
28 cbvrabcsfw.4 . . . . . 6 𝑥𝜓
29 cbvrabcsfw.6 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
3028, 29sbhypf 3500 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
3127, 30anbi12d 633 . . . 4 (𝑧 = 𝑦 → ((𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐵𝜓)))
3218, 19, 31cbvabw 2867 . . 3 {𝑧 ∣ (𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑)} = {𝑦 ∣ (𝑦𝐵𝜓)}
3311, 32eqtri 2821 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦 ∣ (𝑦𝐵𝜓)}
34 df-rab 3115 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
35 df-rab 3115 . 2 {𝑦𝐵𝜓} = {𝑦 ∣ (𝑦𝐵𝜓)}
3633, 34, 353eqtr4i 2831 1 {𝑥𝐴𝜑} = {𝑦𝐵𝜓}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Ⅎwnf 1785  [wsb 2069   ∈ wcel 2111  {cab 2776  Ⅎwnfc 2936  {crab 3110  ⦋csb 3828 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-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  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  df-v 3443  df-sbc 3721  df-csb 3829 This theorem is referenced by:  cbvrabv2w  41827  smfsup  43506  smfinflem  43509  smfinf  43510
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