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Theorem csbprc 3290
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 2910 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 sbcex 2824 . . . . . . 7 ([𝐴 / 𝑥]𝑦𝐵𝐴 ∈ V)
32con3i 595 . . . . . 6 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦𝐵)
43pm2.21d 582 . . . . 5 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝐵 → ⊥))
5 falim 1299 . . . . 5 (⊥ → [𝐴 / 𝑥]𝑦𝐵)
64, 5impbid1 140 . . . 4 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝐵 ↔ ⊥))
76abbidv 2197 . . 3 𝐴 ∈ V → {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦 ∣ ⊥})
8 fal 1292 . . . 4 ¬ ⊥
98abf 3288 . . 3 {𝑦 ∣ ⊥} = ∅
107, 9syl6eq 2130 . 2 𝐴 ∈ V → {𝑦[𝐴 / 𝑥]𝑦𝐵} = ∅)
111, 10syl5eq 2126 1 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1285  wfal 1290  wcel 1434  {cab 2068  Vcvv 2602  [wsbc 2816  csb 2909  c0 3252
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-in 2980  df-ss 2987  df-nul 3253
This theorem is referenced by: (None)
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