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Theorem sbcne12g 3112
Description: Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.)
Assertion
Ref Expression
sbcne12g  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =/=  C  <->  [_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C ) )

Proof of Theorem sbcne12g
StepHypRef Expression
1 nne 2463 . . . . 5  |-  ( -.  B  =/=  C  <->  B  =  C )
21sbcbii 3059 . . . 4  |-  ( [. A  /  x ].  -.  B  =/=  C  <->  [. A  /  x ]. B  =  C )
32a1i 10 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ].  -.  B  =/=  C  <->  [. A  /  x ]. B  =  C )
)
4 sbcng 3044 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ].  -.  B  =/=  C  <->  -. 
[. A  /  x ]. B  =/=  C
) )
5 sbceqg 3110 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
6 nne 2463 . . . 4  |-  ( -. 
[_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C
)
75, 6syl6bbr 254 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  -. 
[_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C ) )
83, 4, 73bitr3d 274 . 2  |-  ( A  e.  V  ->  ( -.  [. A  /  x ]. B  =/=  C  <->  -. 
[_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C ) )
98con4bid 284 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =/=  C  <->  [_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696    =/= wne 2459   [.wsbc 3004   [_csb 3094
This theorem is referenced by:  disjdsct  23384  cdlemkid3N  31744  cdlemkid4  31745
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-sbc 3005  df-csb 3095
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