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Theorem sbceqbid 3409
 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 2728 . . 3 (𝜑 → {𝑥𝜓} = {𝑥𝜒})
41, 3eleq12d 2682 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝐵 ∈ {𝑥𝜒}))
5 df-sbc 3403 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ {𝑥𝜓})
6 df-sbc 3403 . 2 ([𝐵 / 𝑥]𝜒𝐵 ∈ {𝑥𝜒})
74, 5, 63bitr4g 302 1 (𝜑 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑥]𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cab 2596  [wsbc 3402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-sbc 3403 This theorem is referenced by:  fpwwe2cbv  9331  fpwwe2lem2  9333  fpwwe2lem3  9334  fi1uzind  13134  fi1uzindOLD  13140  isprs  16753  isdrs  16757  istos  16858  isdlat  17016  issrg  18330  islmod  18690  fdc  32711  hdmap1ffval  36103  hdmap1fval  36104  hdmapffval  36136  hdmapfval  36137  hgmapffval  36195  hgmapfval  36196  sbccomieg  36375  rexrabdioph  36376
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