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Theorem bj-csbprc 35876
Description: More direct proof of csbprc 4406 (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 3894 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 sbcex 3787 . . . . 5 ([𝐴 / 𝑥]𝑦𝐵𝐴 ∈ V)
32con3i 154 . . . 4 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦𝐵)
43alrimiv 1930 . . 3 𝐴 ∈ V → ∀𝑦 ¬ [𝐴 / 𝑥]𝑦𝐵)
5 bj-ab0 35874 . . 3 (∀𝑦 ¬ [𝐴 / 𝑥]𝑦𝐵 → {𝑦[𝐴 / 𝑥]𝑦𝐵} = ∅)
64, 5syl 17 . 2 𝐴 ∈ V → {𝑦[𝐴 / 𝑥]𝑦𝐵} = ∅)
71, 6eqtrid 2784 1 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1539   = wceq 1541  wcel 2106  {cab 2709  Vcvv 3474  [wsbc 3777  csb 3893  c0 4322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-nul 4323
This theorem is referenced by: (None)
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