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Theorem sbcnel12g 3098
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 2449 . . . 4  |-  ( B  e/  C  <->  -.  B  e.  C )
21sbcbii 3046 . . 3  |-  ( [. A  /  x ]. B  e/  C  <->  [. A  /  x ].  -.  B  e.  C
)
32a1i 10 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e/  C  <->  [. A  /  x ].  -.  B  e.  C ) )
4 sbcng 3031 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ].  -.  B  e.  C  <->  -. 
[. A  /  x ]. B  e.  C
) )
5 sbcel12g 3096 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )
65notbid 285 . . 3  |-  ( A  e.  V  ->  ( -.  [. A  /  x ]. B  e.  C  <->  -. 
[_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )
7 df-nel 2449 . . 3  |-  ( [_ A  /  x ]_ B  e/  [_ A  /  x ]_ C  <->  -.  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C
)
86, 7syl6bbr 254 . 2  |-  ( A  e.  V  ->  ( -.  [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e/  [_ A  /  x ]_ C ) )
93, 4, 83bitrd 270 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 176    e. wcel 1684    e/ wnel 2447   [.wsbc 2991   [_csb 3081
This theorem is referenced by:  rusbcALT  27639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-nel 2449  df-v 2790  df-sbc 2992  df-csb 3082
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