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Theorem bj-csbprc 32714
Description: More direct proof of csbprc 3774 (fewer essential steps). (Contributed by BJ, 24-Jul-2019.)
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 3390 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 sbcex 3297 . . . . 5  |-  ( [. A  /  x ]. y  e.  B  ->  A  e. 
_V )
32con3i 135 . . . 4  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. y  e.  B
)
43alrimiv 1686 . . 3  |-  ( -.  A  e.  _V  ->  A. y  -.  [. A  /  x ]. y  e.  B )
5 bj-abfal 32712 . . 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 2504 1  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1368    = wceq 1370    e. wcel 1758   {cab 2436   _Vcvv 3071   [.wsbc 3287   [_csb 3389   (/)c0 3738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-nul 3739
This theorem is referenced by: (None)
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