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Theorem csbprc 3325
Description: The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.)
Assertion
Ref Expression
csbprc 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)

Proof of Theorem csbprc
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-csb 2932 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 sbcex 2846 . . . . . . 7 ([𝐴 / 𝑥]𝑦𝐵𝐴 ∈ V)
32con3i 597 . . . . . 6 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦𝐵)
43pm2.21d 584 . . . . 5 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝐵 → ⊥))
5 falim 1303 . . . . 5 (⊥ → [𝐴 / 𝑥]𝑦𝐵)
64, 5impbid1 140 . . . 4 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝐵 ↔ ⊥))
76abbidv 2205 . . 3 𝐴 ∈ V → {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦 ∣ ⊥})
8 fal 1296 . . . 4 ¬ ⊥
98abf 3323 . . 3 {𝑦 ∣ ⊥} = ∅
107, 9syl6eq 2136 . 2 𝐴 ∈ V → {𝑦[𝐴 / 𝑥]𝑦𝐵} = ∅)
111, 10syl5eq 2132 1 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1289  wfal 1294  wcel 1438  {cab 2074  Vcvv 2619  [wsbc 2838  csb 2931  c0 3284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-in 3003  df-ss 3010  df-nul 3285
This theorem is referenced by: (None)
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