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Theorem sbceqbid 3583
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 2879 . . 3 (𝜑 → {𝑥𝜓} = {𝑥𝜒})
41, 3eleq12d 2833 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝐵 ∈ {𝑥𝜒}))
5 df-sbc 3577 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ {𝑥𝜓})
6 df-sbc 3577 . 2 ([𝐵 / 𝑥]𝜒𝐵 ∈ {𝑥𝜒})
74, 5, 63bitr4g 303 1 (𝜑 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑥]𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wcel 2139  {cab 2746  [wsbc 3576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-sbc 3577
This theorem is referenced by:  fpwwe2cbv  9644  fpwwe2lem2  9646  fpwwe2lem3  9647  fi1uzind  13471  isprs  17131  isdrs  17135  istos  17236  isdlat  17394  issrg  18707  islmod  19069  fdc  33854  hdmap1ffval  37587  hdmap1fval  37588  hdmapffval  37620  hdmapfval  37621  hgmapffval  37679  hgmapfval  37680  sbccomieg  37859  rexrabdioph  37860
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