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Theorem sbceqbid 3778
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 2885 . . 3 (𝜑 → {𝑥𝜓} = {𝑥𝜒})
41, 3eleq12d 2907 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝐵 ∈ {𝑥𝜒}))
5 df-sbc 3772 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ {𝑥𝜓})
6 df-sbc 3772 . 2 ([𝐵 / 𝑥]𝜒𝐵 ∈ {𝑥𝜒})
74, 5, 63bitr4g 315 1 (𝜑 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑥]𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wcel 2105  {cab 2799  [wsbc 3771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-sbc 3772
This theorem is referenced by:  sbcbidv  3826  fpwwe2cbv  10041  fpwwe2lem2  10043  fpwwe2lem3  10044  fi1uzind  13845  isprs  17530  isdrs  17534  istos  17635  isdlat  17793  issrg  19188  islmod  19569  fdc  34903  hdmap1ffval  38813  hdmap1fval  38814  hdmapffval  38844  hdmapfval  38845  hgmapffval  38903  hgmapfval  38904  sbccomieg  39270  rexrabdioph  39271
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