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Theorem csbing 3540
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 3246 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ ( C  i^i  D )  = 
[_ A  /  x ]_ ( C  i^i  D
) )
2 csbeq1 3246 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
3 csbeq1 3246 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ D  = 
[_ A  /  x ]_ D )
42, 3ineq12d 3535 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D )  =  ( [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D ) )
51, 4eqeq12d 2449 . 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 2951 . . 3  |-  y  e. 
_V
7 nfcsb1v 3275 . . . 4  |-  F/_ x [_ y  /  x ]_ C
8 nfcsb1v 3275 . . . 4  |-  F/_ x [_ y  /  x ]_ D
97, 8nfin 3539 . . 3  |-  F/_ x
( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D )
10 csbeq1a 3251 . . . 4  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
11 csbeq1a 3251 . . . 4  |-  ( x  =  y  ->  D  =  [_ y  /  x ]_ D )
1210, 11ineq12d 3535 . . 3  |-  ( x  =  y  ->  ( C  i^i  D )  =  ( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D ) )
136, 9, 12csbief 3284 . 2  |-  [_ y  /  x ]_ ( C  i^i  D )  =  ( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D )
145, 13vtoclg 3003 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 1652    e. wcel 1725   [_csb 3243    i^i cin 3311
This theorem is referenced by:  csbresg  5141  disjxpin  24020  onfrALTlem5  28555  onfrALTlem4  28556  onfrALTlem5VD  28924  onfrALTlem4VD  28925  csbresgVD  28934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-in 3319
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