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Theorem bj-csbprc 34877
Description: More direct proof of csbprc 3820 (fewer essential steps). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-csbprc  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  (/) )

Proof of Theorem bj-csbprc
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-csb 3421 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 sbcex 3334 . . . . 5  |-  ( [. A  /  x ]. y  e.  B  ->  A  e. 
_V )
32con3i 135 . . . 4  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. y  e.  B
)
43alrimiv 1724 . . 3  |-  ( -.  A  e.  _V  ->  A. y  -.  [. A  /  x ]. y  e.  B )
5 bj-abfal 34875 . . 3  |-  ( A. y  -.  [. A  /  x ]. y  e.  B  ->  { y  |  [. A  /  x ]. y  e.  B }  =  (/) )
64, 5syl 16 . 2  |-  ( -.  A  e.  _V  ->  { y  |  [. A  /  x ]. y  e.  B }  =  (/) )
71, 6syl5eq 2507 1  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1396    = wceq 1398    e. wcel 1823   {cab 2439   _Vcvv 3106   [.wsbc 3324   [_csb 3420   (/)c0 3783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-nul 3784
This theorem is referenced by: (None)
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