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Theorem csbab 4151
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 2747 . . . 4 (𝑧 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
2 sbsbc 3580 . . . 4 ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
31, 2bitri 264 . . 3 (𝑧 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
4 sbccom 3650 . . . 4 ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
5 df-clab 2747 . . . . . 6 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
6 sbsbc 3580 . . . . . 6 ([𝑧 / 𝑦]𝜑[𝑧 / 𝑦]𝜑)
75, 6bitri 264 . . . . 5 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
87sbcbii 3632 . . . 4 ([𝐴 / 𝑥]𝑧 ∈ {𝑦𝜑} ↔ [𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
94, 8bitr4i 267 . . 3 ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝑧 ∈ {𝑦𝜑})
10 sbcel2 4132 . . 3 ([𝐴 / 𝑥]𝑧 ∈ {𝑦𝜑} ↔ 𝑧𝐴 / 𝑥{𝑦𝜑})
113, 9, 103bitrri 287 . 2 (𝑧𝐴 / 𝑥{𝑦𝜑} ↔ 𝑧 ∈ {𝑦[𝐴 / 𝑥]𝜑})
1211eqriv 2757 1 𝐴 / 𝑥{𝑦𝜑} = {𝑦[𝐴 / 𝑥]𝜑}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  [wsb 2046  wcel 2139  {cab 2746  [wsbc 3576  csb 3674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-nul 4059
This theorem is referenced by:  csbsng  4387  csbuni  4618  csbxp  5357  csbdm  5473  csbwrdg  13520  abfmpeld  29763  abfmpel  29764  csbwrecsg  33484  csboprabg  33487  csbfinxpg  33536  csbxpgOLD  39553  csbrngOLD  39556  csbfv12gALTVD  39634
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