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Theorem sbcnel12g 3773
Description: Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)
Assertion
Ref Expression
sbcnel12g  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e/  C  <->  [_ A  /  x ]_ B  e/  [_ A  /  x ]_ C ) )

Proof of Theorem sbcnel12g
StepHypRef Expression
1 sbcng 3307 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ].  -.  B  e.  C  <->  -. 
[. A  /  x ]. B  e.  C
) )
2 df-nel 2624 . . 3  |-  ( B  e/  C  <->  -.  B  e.  C )
32sbcbii 3322 . 2  |-  ( [. A  /  x ]. B  e/  C  <->  [. A  /  x ].  -.  B  e.  C
)
4 df-nel 2624 . . 3  |-  ( [_ A  /  x ]_ B  e/  [_ A  /  x ]_ C  <->  -.  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C
)
5 sbcel12 3771 . . 3  |-  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C )
64, 5xchbinxr 313 . 2  |-  ( [_ A  /  x ]_ B  e/  [_ A  /  x ]_ C  <->  -.  [. A  /  x ]. B  e.  C
)
71, 3, 63bitr4g 292 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e/  C  <->  [_ A  /  x ]_ B  e/  [_ A  /  x ]_ C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    e. wcel 1886    e/ wnel 2622   [.wsbc 3266   [_csb 3362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-nel 2624  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-in 3410  df-ss 3417  df-nul 3731
This theorem is referenced by:  rusbcALT  36784
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