MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcnel12g Unicode version

Theorem sbcnel12g 3211
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 2553 . . . 4  |-  ( B  e/  C  <->  -.  B  e.  C )
21sbcbii 3159 . . 3  |-  ( [. A  /  x ]. B  e/  C  <->  [. A  /  x ].  -.  B  e.  C
)
32a1i 11 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e/  C  <->  [. A  /  x ].  -.  B  e.  C ) )
4 sbcng 3144 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ].  -.  B  e.  C  <->  -. 
[. A  /  x ]. B  e.  C
) )
5 sbcel12g 3209 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )
65notbid 286 . . 3  |-  ( A  e.  V  ->  ( -.  [. A  /  x ]. B  e.  C  <->  -. 
[_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )
7 df-nel 2553 . . 3  |-  ( [_ A  /  x ]_ B  e/  [_ A  /  x ]_ C  <->  -.  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C
)
86, 7syl6bbr 255 . 2  |-  ( A  e.  V  ->  ( -.  [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e/  [_ A  /  x ]_ C ) )
93, 4, 83bitrd 271 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 177    e. wcel 1717    e/ wnel 2551   [.wsbc 3104   [_csb 3194
This theorem is referenced by:  rusbcALT  27308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-nel 2553  df-v 2901  df-sbc 3105  df-csb 3195
  Copyright terms: Public domain W3C validator