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Theorem csbab 3960
 Description: Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Revised by NM, 19-Aug-2018.)
Assertion
Ref Expression
csbab 𝐴 / 𝑥{𝑦𝜑} = {𝑦[𝐴 / 𝑥]𝜑}
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem csbab
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-clab 2597 . . . 4 (𝑧 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
2 sbsbc 3406 . . . 4 ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
31, 2bitri 263 . . 3 (𝑧 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
4 sbccom 3476 . . . 4 ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
5 df-clab 2597 . . . . . 6 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
6 sbsbc 3406 . . . . . 6 ([𝑧 / 𝑦]𝜑[𝑧 / 𝑦]𝜑)
75, 6bitri 263 . . . . 5 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
87sbcbii 3458 . . . 4 ([𝐴 / 𝑥]𝑧 ∈ {𝑦𝜑} ↔ [𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
94, 8bitr4i 266 . . 3 ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝑧 ∈ {𝑦𝜑})
10 sbcel2 3941 . . 3 ([𝐴 / 𝑥]𝑧 ∈ {𝑦𝜑} ↔ 𝑧𝐴 / 𝑥{𝑦𝜑})
113, 9, 103bitrri 286 . 2 (𝑧𝐴 / 𝑥{𝑦𝜑} ↔ 𝑧 ∈ {𝑦[𝐴 / 𝑥]𝜑})
1211eqriv 2607 1 𝐴 / 𝑥{𝑦𝜑} = {𝑦[𝐴 / 𝑥]𝜑}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  [wsb 1867   ∈ wcel 1977  {cab 2596  [wsbc 3402  ⦋csb 3499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-nul 3875 This theorem is referenced by:  csbsng  4190  csbuni  4402  csbxp  5123  csbdm  5240  csbwrdg  13189  abfmpeld  28834  abfmpel  28835  csbwrecsg  32349  csboprabg  32352  csbfinxpg  32401  csbfv12gALTOLD  38074  csbfv12gALTVD  38157
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