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Theorem csbidm 4330
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 4329 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐴 / 𝑥𝐵)
2 csbconstg 3819 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐴 = 𝐴)
32csbeq1d 3804 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
41, 3eqtrd 2774 . 2 (𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
5 csbprc 4305 . . 3 𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = ∅)
6 csbprc 4305 . . 3 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
75, 6eqtr4d 2777 . 2 𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
84, 7pm2.61i 185 1 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2114  Vcvv 3400  csb 3800  c0 4221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-nul 4222
This theorem is referenced by: (None)
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