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Theorem sbcnel12g 3803
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 3341 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ].  -.  B  e.  C  <->  -. 
[. A  /  x ]. B  e.  C
) )
2 df-nel 2622 . . 3  |-  ( B  e/  C  <->  -.  B  e.  C )
32sbcbii 3356 . 2  |-  ( [. A  /  x ]. B  e/  C  <->  [. A  /  x ].  -.  B  e.  C
)
4 df-nel 2622 . . 3  |-  ( [_ A  /  x ]_ B  e/  [_ A  /  x ]_ C  <->  -.  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C
)
5 sbcel12 3801 . . 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 1869    e/ wnel 2620   [.wsbc 3300   [_csb 3396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-nel 2622  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-in 3444  df-ss 3451  df-nul 3763
This theorem is referenced by:  rusbcALT  36654
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