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Theorem bj-csbprc 33801
Description: More direct proof of csbprc 4278 (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 3812 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 sbcex 3716 . . . . 5 ([𝐴 / 𝑥]𝑦𝐵𝐴 ∈ V)
32con3i 157 . . . 4 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦𝐵)
43alrimiv 1905 . . 3 𝐴 ∈ V → ∀𝑦 ¬ [𝐴 / 𝑥]𝑦𝐵)
5 bj-ab0 33799 . . 3 (∀𝑦 ¬ [𝐴 / 𝑥]𝑦𝐵 → {𝑦[𝐴 / 𝑥]𝑦𝐵} = ∅)
64, 5syl 17 . 2 𝐴 ∈ V → {𝑦[𝐴 / 𝑥]𝑦𝐵} = ∅)
71, 6syl5eq 2843 1 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1520   = wceq 1522  wcel 2081  {cab 2775  Vcvv 3437  [wsbc 3706  csb 3811  c0 4211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-nul 4212
This theorem is referenced by: (None)
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