MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbceqbid Structured version   Visualization version   GIF version

Theorem sbceqbid 3785
Description: Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.)
Hypotheses
Ref Expression
sbceqbid.1 (𝜑𝐴 = 𝐵)
sbceqbid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
sbceqbid (𝜑 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑥]𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem sbceqbid
StepHypRef Expression
1 sbceqbid.1 . . 3 (𝜑𝐴 = 𝐵)
2 sbceqbid.2 . . . 4 (𝜑 → (𝜓𝜒))
32abbidv 2802 . . 3 (𝜑 → {𝑥𝜓} = {𝑥𝜒})
41, 3eleq12d 2828 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝐵 ∈ {𝑥𝜒}))
5 df-sbc 3779 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ {𝑥𝜓})
6 df-sbc 3779 . 2 ([𝐵 / 𝑥]𝜒𝐵 ∈ {𝑥𝜒})
74, 5, 63bitr4g 314 1 (𝜑 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑥]𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  {cab 2710  [wsbc 3778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-sbc 3779
This theorem is referenced by:  sbcbidv  3837  frpoins3xpg  8126  frpoins3xp3g  8127  fpwwe2cbv  10625  fpwwe2lem2  10627  fpwwe2lem3  10628  fi1uzind  14458  isprs  18250  isdrs  18254  istos  18371  isdlat  18475  issrg  20011  islmod  20475  fdc  36613  hdmap1ffval  40666  hdmap1fval  40667  hdmapffval  40697  hdmapfval  40698  hgmapffval  40756  hgmapfval  40757  sbccomieg  41531  rexrabdioph  41532
  Copyright terms: Public domain W3C validator