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Theorem cbvrabcsfw 3855
Description: Version of cbvrabcsf 3859 with a disjoint variable condition, which does not require ax-13 2371. (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 1922 . . . 4 𝑧(𝑥𝐴𝜑)
2 nfcsb1v 3836 . . . . . 6 𝑥𝑧 / 𝑥𝐴
32nfcri 2891 . . . . 5 𝑥 𝑧𝑧 / 𝑥𝐴
4 nfs1v 2157 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
53, 4nfan 1907 . . . 4 𝑥(𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑)
6 id 22 . . . . . 6 (𝑥 = 𝑧𝑥 = 𝑧)
7 csbeq1a 3825 . . . . . 6 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
86, 7eleq12d 2832 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝑧 / 𝑥𝐴))
9 sbequ12 2249 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
108, 9anbi12d 634 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝜑) ↔ (𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑)))
111, 5, 10cbvabw 2812 . . 3 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑧 ∣ (𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑)}
12 nfcv 2904 . . . . . . 7 𝑦𝑧
13 cbvrabcsfw.1 . . . . . . 7 𝑦𝐴
1412, 13nfcsbw 3838 . . . . . 6 𝑦𝑧 / 𝑥𝐴
1514nfcri 2891 . . . . 5 𝑦 𝑧𝑧 / 𝑥𝐴
16 cbvrabcsfw.3 . . . . . 6 𝑦𝜑
1716nfsbv 2329 . . . . 5 𝑦[𝑧 / 𝑥]𝜑
1815, 17nfan 1907 . . . 4 𝑦(𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑)
19 nfv 1922 . . . 4 𝑧(𝑦𝐵𝜓)
20 id 22 . . . . . 6 (𝑧 = 𝑦𝑧 = 𝑦)
21 csbeq1 3814 . . . . . . 7 (𝑧 = 𝑦𝑧 / 𝑥𝐴 = 𝑦 / 𝑥𝐴)
22 vex 3412 . . . . . . . 8 𝑦 ∈ V
23 cbvrabcsfw.2 . . . . . . . 8 𝑥𝐵
24 cbvrabcsfw.5 . . . . . . . 8 (𝑥 = 𝑦𝐴 = 𝐵)
2522, 23, 24csbief 3846 . . . . . . 7 𝑦 / 𝑥𝐴 = 𝐵
2621, 25eqtrdi 2794 . . . . . 6 (𝑧 = 𝑦𝑧 / 𝑥𝐴 = 𝐵)
2720, 26eleq12d 2832 . . . . 5 (𝑧 = 𝑦 → (𝑧𝑧 / 𝑥𝐴𝑦𝐵))
28 cbvrabcsfw.4 . . . . . 6 𝑥𝜓
29 cbvrabcsfw.6 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
3028, 29sbhypf 3467 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
3127, 30anbi12d 634 . . . 4 (𝑧 = 𝑦 → ((𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐵𝜓)))
3218, 19, 31cbvabw 2812 . . 3 {𝑧 ∣ (𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑)} = {𝑦 ∣ (𝑦𝐵𝜓)}
3311, 32eqtri 2765 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦 ∣ (𝑦𝐵𝜓)}
34 df-rab 3070 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
35 df-rab 3070 . 2 {𝑦𝐵𝜓} = {𝑦 ∣ (𝑦𝐵𝜓)}
3633, 34, 353eqtr4i 2775 1 {𝑥𝐴𝜑} = {𝑦𝐵𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wnf 1791  [wsb 2070  wcel 2110  {cab 2714  wnfc 2884  {crab 3065  csb 3811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812
This theorem is referenced by:  cbvrabv2w  42350  smfsup  44019  smfinflem  44022  smfinf  44023
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