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Theorem bj-csbprc 33210
Description: More direct proof of csbprc 4178 (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 3729 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 sbcex 3643 . . . . 5 ([𝐴 / 𝑥]𝑦𝐵𝐴 ∈ V)
32con3i 151 . . . 4 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦𝐵)
43alrimiv 2018 . . 3 𝐴 ∈ V → ∀𝑦 ¬ [𝐴 / 𝑥]𝑦𝐵)
5 bj-ab0 33208 . . 3 (∀𝑦 ¬ [𝐴 / 𝑥]𝑦𝐵 → {𝑦[𝐴 / 𝑥]𝑦𝐵} = ∅)
64, 5syl 17 . 2 𝐴 ∈ V → {𝑦[𝐴 / 𝑥]𝑦𝐵} = ∅)
71, 6syl5eq 2852 1 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1635   = wceq 1637  wcel 2156  {cab 2792  Vcvv 3391  [wsbc 3633  csb 3728  c0 4116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-nul 4117
This theorem is referenced by: (None)
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