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Theorem csbab 4438
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 2711 . . . 4 (𝑧 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
2 sbsbc 3782 . . . 4 ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
31, 2bitri 275 . . 3 (𝑧 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
4 sbccom 3866 . . . 4 ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
5 df-clab 2711 . . . . . 6 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
6 sbsbc 3782 . . . . . 6 ([𝑧 / 𝑦]𝜑[𝑧 / 𝑦]𝜑)
75, 6bitri 275 . . . . 5 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
87sbcbii 3838 . . . 4 ([𝐴 / 𝑥]𝑧 ∈ {𝑦𝜑} ↔ [𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
94, 8bitr4i 278 . . 3 ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝑧 ∈ {𝑦𝜑})
10 sbcel2 4416 . . 3 ([𝐴 / 𝑥]𝑧 ∈ {𝑦𝜑} ↔ 𝑧𝐴 / 𝑥{𝑦𝜑})
113, 9, 103bitrri 298 . 2 (𝑧𝐴 / 𝑥{𝑦𝜑} ↔ 𝑧 ∈ {𝑦[𝐴 / 𝑥]𝜑})
1211eqriv 2730 1 𝐴 / 𝑥{𝑦𝜑} = {𝑦[𝐴 / 𝑥]𝜑}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  [wsb 2068  wcel 2107  {cab 2710  [wsbc 3778  csb 3894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-nul 4324
This theorem is referenced by:  csbsng  4713  csbuni  4941  csbxp  5776  csbdm  5898  csbfrecsg  8269  csbwrdg  14494  abfmpeld  31879  abfmpel  31880  csboprabg  36211  csbfinxpg  36269  csbingVD  43645  csbsngVD  43654  csbxpgVD  43655  csbrngVD  43657  csbunigVD  43659  csbfv12gALTVD  43660
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