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Theorem csbidm 4146
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 4145 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐴 / 𝑥𝐵)
2 csbconstg 3695 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐴 = 𝐴)
32csbeq1d 3689 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
41, 3eqtrd 2805 . 2 (𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
5 csbprc 4124 . . 3 𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = ∅)
6 csbprc 4124 . . 3 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
75, 6eqtr4d 2808 . 2 𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
84, 7pm2.61i 176 1 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1631  wcel 2145  Vcvv 3351  csb 3682  c0 4063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-nul 4064
This theorem is referenced by: (None)
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