MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbprcOLD Structured version   Visualization version   GIF version

Theorem csbprcOLD 4057
Description: Obsolete proof of csbprc 4056 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 3608 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 sbcex 3519 . . . . . . 7 ([𝐴 / 𝑥]𝑦𝐵𝐴 ∈ V)
32con3i 150 . . . . . 6 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦𝐵)
43pm2.21d 118 . . . . 5 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝐵 → ⊥))
5 falim 1579 . . . . 5 (⊥ → [𝐴 / 𝑥]𝑦𝐵)
64, 5impbid1 215 . . . 4 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝐵 ↔ ⊥))
76abbidv 2811 . . 3 𝐴 ∈ V → {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦 ∣ ⊥})
8 fal 1571 . . . 4 ¬ ⊥
98abf 4054 . . 3 {𝑦 ∣ ⊥} = ∅
107, 9syl6eq 2742 . 2 𝐴 ∈ V → {𝑦[𝐴 / 𝑥]𝑦𝐵} = ∅)
111, 10syl5eq 2738 1 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1564  wfal 1569  wcel 2071  {cab 2678  Vcvv 3272  [wsbc 3509  csb 3607  c0 3991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1567  df-fal 1570  df-ex 1786  df-nf 1791  df-sb 1979  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-v 3274  df-sbc 3510  df-csb 3608  df-dif 3651  df-nul 3992
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator