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Theorem bj-csbprc 34632
 Description: More direct proof of csbprc 4303 (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 3807 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 sbcex 3707 . . . . 5 ([𝐴 / 𝑥]𝑦𝐵𝐴 ∈ V)
32con3i 157 . . . 4 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦𝐵)
43alrimiv 1929 . . 3 𝐴 ∈ V → ∀𝑦 ¬ [𝐴 / 𝑥]𝑦𝐵)
5 bj-ab0 34630 . . 3 (∀𝑦 ¬ [𝐴 / 𝑥]𝑦𝐵 → {𝑦[𝐴 / 𝑥]𝑦𝐵} = ∅)
64, 5syl 17 . 2 𝐴 ∈ V → {𝑦[𝐴 / 𝑥]𝑦𝐵} = ∅)
71, 6syl5eq 2806 1 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537   = wceq 1539   ∈ wcel 2112  {cab 2736  Vcvv 3410  [wsbc 3697  ⦋csb 3806  ∅c0 4226 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-nul 4227 This theorem is referenced by: (None)
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