ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbcel1v GIF version

Theorem sbcel1v 2971
Description: Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.)
Assertion
Ref Expression
sbcel1v ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵)
Distinct variable group:   𝑥,𝐵
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem sbcel1v
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sbcex 2917 . 2 ([𝐴 / 𝑥]𝑥𝐵𝐴 ∈ V)
2 elex 2697 . 2 (𝐴𝐵𝐴 ∈ V)
3 dfsbcq2 2912 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥𝐵[𝐴 / 𝑥]𝑥𝐵))
4 eleq1 2202 . . 3 (𝑦 = 𝐴 → (𝑦𝐵𝐴𝐵))
5 clelsb3 2244 . . 3 ([𝑦 / 𝑥]𝑥𝐵𝑦𝐵)
63, 4, 5vtoclbg 2747 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵))
71, 2, 6pm5.21nii 693 1 ([𝐴 / 𝑥]𝑥𝐵𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wb 104  wcel 1480  [wsb 1735  Vcvv 2686  [wsbc 2909
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910
This theorem is referenced by:  f1od2  6132
  Copyright terms: Public domain W3C validator