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Theorem bj-csbprc 32096
 Description: More direct proof of csbprc 3932 (fewer essential steps). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-csbprc 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)

Proof of Theorem bj-csbprc
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-csb 3500 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 sbcex 3412 . . . . 5 ([𝐴 / 𝑥]𝑦𝐵𝐴 ∈ V)
32con3i 149 . . . 4 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦𝐵)
43alrimiv 1842 . . 3 𝐴 ∈ V → ∀𝑦 ¬ [𝐴 / 𝑥]𝑦𝐵)
5 bj-ab0 32094 . . 3 (∀𝑦 ¬ [𝐴 / 𝑥]𝑦𝐵 → {𝑦[𝐴 / 𝑥]𝑦𝐵} = ∅)
64, 5syl 17 . 2 𝐴 ∈ V → {𝑦[𝐴 / 𝑥]𝑦𝐵} = ∅)
71, 6syl5eq 2656 1 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173  [wsbc 3402  ⦋csb 3499  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-nul 3875 This theorem is referenced by: (None)
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