MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbidm Structured version   Visualization version   GIF version

Theorem csbidm 4197
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 4196 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐴 / 𝑥𝐵)
2 csbconstg 3739 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐴 = 𝐴)
32csbeq1d 3733 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
41, 3eqtrd 2838 . 2 (𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
5 csbprc 4176 . . 3 𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = ∅)
6 csbprc 4176 . . 3 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
75, 6eqtr4d 2841 . 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 1637  wcel 2156  Vcvv 3389  csb 3726  c0 4114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2782
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-v 3391  df-sbc 3632  df-csb 3727  df-dif 3770  df-nul 4115
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator