ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbccsbg GIF version

Theorem sbccsbg 2957
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 2076 . . 3 (𝑦 ∈ {𝑦𝜑} ↔ 𝜑)
21sbcbii 2896 . 2 ([𝐴 / 𝑥]𝑦 ∈ {𝑦𝜑} ↔ [𝐴 / 𝑥]𝜑)
3 sbcel2g 2950 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦 ∈ {𝑦𝜑} ↔ 𝑦𝐴 / 𝑥{𝑦𝜑}))
42, 3syl5bbr 192 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝑦𝐴 / 𝑥{𝑦𝜑}))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wcel 1438  {cab 2074  [wsbc 2838  csb 2931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2839  df-csb 2932
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator