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Theorem csbprc 3262
 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 2853 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 sbcex 2772 . . . . . . 7 ([𝐴 / 𝑥]𝑦𝐵𝐴 ∈ V)
32con3i 562 . . . . . 6 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦𝐵)
43pm2.21d 549 . . . . 5 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝐵 → ⊥))
5 falim 1257 . . . . 5 (⊥ → [𝐴 / 𝑥]𝑦𝐵)
64, 5impbid1 130 . . . 4 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝐵 ↔ ⊥))
76abbidv 2155 . . 3 𝐴 ∈ V → {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦 ∣ ⊥})
8 fal 1250 . . . 4 ¬ ⊥
98abf 3260 . . 3 {𝑦 ∣ ⊥} = ∅
107, 9syl6eq 2088 . 2 𝐴 ∈ V → {𝑦[𝐴 / 𝑥]𝑦𝐵} = ∅)
111, 10syl5eq 2084 1 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1243  ⊥wfal 1248   ∈ wcel 1393  {cab 2026  Vcvv 2557  [wsbc 2764  ⦋csb 2852  ∅c0 3224 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225 This theorem is referenced by: (None)
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