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Theorem csbprc 3378
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 2976 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 sbcex 2890 . . . . . . 7 ([𝐴 / 𝑥]𝑦𝐵𝐴 ∈ V)
32con3i 606 . . . . . 6 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦𝐵)
43pm2.21d 593 . . . . 5 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝐵 → ⊥))
5 falim 1330 . . . . 5 (⊥ → [𝐴 / 𝑥]𝑦𝐵)
64, 5impbid1 141 . . . 4 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝐵 ↔ ⊥))
76abbidv 2235 . . 3 𝐴 ∈ V → {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦 ∣ ⊥})
8 fal 1323 . . . 4 ¬ ⊥
98abf 3376 . . 3 {𝑦 ∣ ⊥} = ∅
107, 9syl6eq 2166 . 2 𝐴 ∈ V → {𝑦[𝐴 / 𝑥]𝑦𝐵} = ∅)
111, 10syl5eq 2162 1 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1316  wfal 1321  wcel 1465  {cab 2103  Vcvv 2660  [wsbc 2882  csb 2975  c0 3333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-in 3047  df-ss 3054  df-nul 3334
This theorem is referenced by: (None)
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