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Theorem csbidm 4379
Description: Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
csbidm 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem csbidm
StepHypRef Expression
1 csbnest1g 4378 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐴 / 𝑥𝐵)
2 csbconstg 3899 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐴 = 𝐴)
32csbeq1d 3884 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
41, 3eqtrd 2853 . 2 (𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
5 csbprc 4355 . . 3 𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = ∅)
6 csbprc 4355 . . 3 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
75, 6eqtr4d 2856 . 2 𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
84, 7pm2.61i 183 1 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1528  wcel 2105  Vcvv 3492  csb 3880  c0 4288
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-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-nul 4289
This theorem is referenced by: (None)
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