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Theorem sbcnel12g 2948
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 df-nel 2351 . . . 4  |-  ( B  e/  C  <->  -.  B  e.  C )
21sbcbii 2898 . . 3  |-  ( [. A  /  x ]. B  e/  C  <->  [. A  /  x ].  -.  B  e.  C
)
32a1i 9 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e/  C  <->  [. A  /  x ].  -.  B  e.  C ) )
4 sbcng 2879 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ].  -.  B  e.  C  <->  -. 
[. A  /  x ]. B  e.  C
) )
5 sbcel12g 2946 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )
65notbid 627 . . 3  |-  ( A  e.  V  ->  ( -.  [. A  /  x ]. B  e.  C  <->  -. 
[_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )
7 df-nel 2351 . . 3  |-  ( [_ A  /  x ]_ B  e/  [_ A  /  x ]_ C  <->  -.  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C
)
86, 7syl6bbr 196 . 2  |-  ( A  e.  V  ->  ( -.  [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e/  [_ A  /  x ]_ C ) )
93, 4, 83bitrd 212 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e/  C  <->  [_ A  /  x ]_ B  e/  [_ A  /  x ]_ C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    e. wcel 1438    e/ wnel 2350   [.wsbc 2840   [_csb 2933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-nel 2351  df-v 2621  df-sbc 2841  df-csb 2934
This theorem is referenced by: (None)
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