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Theorem csbabg 3110
Description: Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
csbabg (𝐴𝑉𝐴 / 𝑥{𝑦𝜑} = {𝑦[𝐴 / 𝑥]𝜑})
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem csbabg
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sbccom 3030 . . . 4 ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
2 df-clab 2157 . . . . 5 (𝑧 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
3 sbsbc 2959 . . . . 5 ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
42, 3bitri 183 . . . 4 (𝑧 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
5 df-clab 2157 . . . . . 6 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
6 sbsbc 2959 . . . . . 6 ([𝑧 / 𝑦]𝜑[𝑧 / 𝑦]𝜑)
75, 6bitri 183 . . . . 5 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
87sbcbii 3014 . . . 4 ([𝐴 / 𝑥]𝑧 ∈ {𝑦𝜑} ↔ [𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
91, 4, 83bitr4i 211 . . 3 (𝑧 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ [𝐴 / 𝑥]𝑧 ∈ {𝑦𝜑})
10 sbcel2g 3070 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧 ∈ {𝑦𝜑} ↔ 𝑧𝐴 / 𝑥{𝑦𝜑}))
119, 10bitr2id 192 . 2 (𝐴𝑉 → (𝑧𝐴 / 𝑥{𝑦𝜑} ↔ 𝑧 ∈ {𝑦[𝐴 / 𝑥]𝜑}))
1211eqrdv 2168 1 (𝐴𝑉𝐴 / 𝑥{𝑦𝜑} = {𝑦[𝐴 / 𝑥]𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  [wsb 1755  wcel 2141  {cab 2156  [wsbc 2955  csb 3049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050
This theorem is referenced by:  csbsng  3644  csbunig  3804  csbxpg  4692  csbdmg  4805  csbrng  5072
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