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Theorem sbccsbg 3101
Description: Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.)
Assertion
Ref Expression
sbccsbg (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝑦𝐴 / 𝑥{𝑦𝜑}))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem sbccsbg
StepHypRef Expression
1 abid 2177 . . 3 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
21sbcbii 3037 . 2 ([𝐴 / 𝑥]𝑦 ∈ {𝑦𝜑} ↔ [𝐴 / 𝑥]𝜑)
3 sbcel2g 3093 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦 ∈ {𝑦𝜑} ↔ 𝑦𝐴 / 𝑥{𝑦𝜑}))
42, 3bitr3id 194 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝑦𝐴 / 𝑥{𝑦𝜑}))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wcel 2160  {cab 2175  [wsbc 2977  csb 3072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978  df-csb 3073
This theorem is referenced by: (None)
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