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Theorem csbing 3389
Description: Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)
Assertion
Ref Expression
csbing  |-  ( A  e.  B  ->  [_ A  /  x ]_ ( C  i^i  D )  =  ( [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D ) )

Proof of Theorem csbing
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3097 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ ( C  i^i  D )  = 
[_ A  /  x ]_ ( C  i^i  D
) )
2 csbeq1 3097 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
3 csbeq1 3097 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ D  = 
[_ A  /  x ]_ D )
42, 3ineq12d 3384 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D )  =  ( [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D ) )
51, 4eqeq12d 2310 . 2  |-  ( y  =  A  ->  ( [_ y  /  x ]_ ( C  i^i  D
)  =  ( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D )  <->  [_ A  /  x ]_ ( C  i^i  D )  =  ( [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D ) ) )
6 vex 2804 . . 3  |-  y  e. 
_V
7 nfcsb1v 3126 . . . 4  |-  F/_ x [_ y  /  x ]_ C
8 nfcsb1v 3126 . . . 4  |-  F/_ x [_ y  /  x ]_ D
97, 8nfin 3388 . . 3  |-  F/_ x
( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D )
10 csbeq1a 3102 . . . 4  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
11 csbeq1a 3102 . . . 4  |-  ( x  =  y  ->  D  =  [_ y  /  x ]_ D )
1210, 11ineq12d 3384 . . 3  |-  ( x  =  y  ->  ( C  i^i  D )  =  ( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D ) )
136, 9, 12csbief 3135 . 2  |-  [_ y  /  x ]_ ( C  i^i  D )  =  ( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D )
145, 13vtoclg 2856 1  |-  ( A  e.  B  ->  [_ A  /  x ]_ ( C  i^i  D )  =  ( [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   [_csb 3094    i^i cin 3164
This theorem is referenced by:  csbresg  4974  onfrALTlem5  28606  onfrALTlem4  28607  onfrALTlem5VD  28977  onfrALTlem4VD  28978  csbresgVD  28987
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-3an 936  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-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-in 3172
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