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Theorem csbprcOLD 3958
Description: Obsolete proof of csbprc 3957 as of 27-Aug-2021. (Contributed by NM, 17-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
csbprcOLD 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)

Proof of Theorem csbprcOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-csb 3520 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 sbcex 3432 . . . . . . 7 ([𝐴 / 𝑥]𝑦𝐵𝐴 ∈ V)
32con3i 150 . . . . . 6 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦𝐵)
43pm2.21d 118 . . . . 5 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝐵 → ⊥))
5 falim 1495 . . . . 5 (⊥ → [𝐴 / 𝑥]𝑦𝐵)
64, 5impbid1 215 . . . 4 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝐵 ↔ ⊥))
76abbidv 2744 . . 3 𝐴 ∈ V → {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦 ∣ ⊥})
8 fal 1487 . . . 4 ¬ ⊥
98abf 3955 . . 3 {𝑦 ∣ ⊥} = ∅
107, 9syl6eq 2676 . 2 𝐴 ∈ V → {𝑦[𝐴 / 𝑥]𝑦𝐵} = ∅)
111, 10syl5eq 2672 1 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480  wfal 1485  wcel 1992  {cab 2612  Vcvv 3191  [wsbc 3422  csb 3519  c0 3896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-nul 3897
This theorem is referenced by: (None)
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