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Theorem csbabg 3092
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 3012 . . . 4 ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
2 df-clab 2144 . . . . 5 (𝑧 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑)
3 sbsbc 2941 . . . . 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 2144 . . . . . 6 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
6 sbsbc 2941 . . . . . 6 ([𝑧 / 𝑦]𝜑[𝑧 / 𝑦]𝜑)
75, 6bitri 183 . . . . 5 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
87sbcbii 2996 . . . 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 3052 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧 ∈ {𝑦𝜑} ↔ 𝑧𝐴 / 𝑥{𝑦𝜑}))
119, 10syl5rbb 192 . 2 (𝐴𝑉 → (𝑧𝐴 / 𝑥{𝑦𝜑} ↔ 𝑧 ∈ {𝑦[𝐴 / 𝑥]𝜑}))
1211eqrdv 2155 1 (𝐴𝑉𝐴 / 𝑥{𝑦𝜑} = {𝑦[𝐴 / 𝑥]𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1335  [wsb 1742  wcel 2128  {cab 2143  [wsbc 2937  csb 3031
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-sbc 2938  df-csb 3032
This theorem is referenced by:  csbsng  3620  csbunig  3780  csbxpg  4667  csbdmg  4780  csbrng  5047
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