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Theorem csbing 3491
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 3197 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ ( C  i^i  D )  = 
[_ A  /  x ]_ ( C  i^i  D
) )
2 csbeq1 3197 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
3 csbeq1 3197 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ D  = 
[_ A  /  x ]_ D )
42, 3ineq12d 3486 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D )  =  ( [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D ) )
51, 4eqeq12d 2401 . 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 2902 . . 3  |-  y  e. 
_V
7 nfcsb1v 3226 . . . 4  |-  F/_ x [_ y  /  x ]_ C
8 nfcsb1v 3226 . . . 4  |-  F/_ x [_ y  /  x ]_ D
97, 8nfin 3490 . . 3  |-  F/_ x
( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D )
10 csbeq1a 3202 . . . 4  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
11 csbeq1a 3202 . . . 4  |-  ( x  =  y  ->  D  =  [_ y  /  x ]_ D )
1210, 11ineq12d 3486 . . 3  |-  ( x  =  y  ->  ( C  i^i  D )  =  ( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D ) )
136, 9, 12csbief 3235 . 2  |-  [_ y  /  x ]_ ( C  i^i  D )  =  ( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D )
145, 13vtoclg 2954 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 1649    e. wcel 1717   [_csb 3194    i^i cin 3262
This theorem is referenced by:  csbresg  5089  onfrALTlem5  27971  onfrALTlem4  27972  onfrALTlem5VD  28338  onfrALTlem4VD  28339  csbresgVD  28348
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-3an 938  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-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-in 3270
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