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Theorem csbing 3144
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 2855 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ ( C  i^i  D )  = 
[_ A  /  x ]_ ( C  i^i  D
) )
2 csbeq1 2855 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
3 csbeq1 2855 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ D  = 
[_ A  /  x ]_ D )
42, 3ineq12d 3139 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D )  =  ( [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D ) )
51, 4eqeq12d 2054 . 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 2560 . . 3  |-  y  e. 
_V
7 nfcsb1v 2882 . . . 4  |-  F/_ x [_ y  /  x ]_ C
8 nfcsb1v 2882 . . . 4  |-  F/_ x [_ y  /  x ]_ D
97, 8nfin 3143 . . 3  |-  F/_ x
( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D )
10 csbeq1a 2860 . . . 4  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
11 csbeq1a 2860 . . . 4  |-  ( x  =  y  ->  D  =  [_ y  /  x ]_ D )
1210, 11ineq12d 3139 . . 3  |-  ( x  =  y  ->  ( C  i^i  D )  =  ( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D ) )
136, 9, 12csbief 2891 . 2  |-  [_ y  /  x ]_ ( C  i^i  D )  =  ( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D )
145, 13vtoclg 2613 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 1243    e. wcel 1393   [_csb 2852    i^i cin 2916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-in 2924
This theorem is referenced by:  csbresg  4615
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