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Theorem csbingOLD 37254
Description: Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.) Obsolete as of 18-Aug-2018. Use csbin 3810 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
csbingOLD  |-  ( A  e.  B  ->  [_ A  /  x ]_ ( C  i^i  D )  =  ( [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D ) )

Proof of Theorem csbingOLD
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3377 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ ( C  i^i  D )  = 
[_ A  /  x ]_ ( C  i^i  D
) )
2 csbeq1 3377 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
3 csbeq1 3377 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ D  = 
[_ A  /  x ]_ D )
42, 3ineq12d 3646 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D )  =  ( [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D ) )
51, 4eqeq12d 2476 . 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 3059 . . 3  |-  y  e. 
_V
7 nfcsb1v 3390 . . . 4  |-  F/_ x [_ y  /  x ]_ C
8 nfcsb1v 3390 . . . 4  |-  F/_ x [_ y  /  x ]_ D
97, 8nfin 3650 . . 3  |-  F/_ x
( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D )
10 csbeq1a 3383 . . . 4  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
11 csbeq1a 3383 . . . 4  |-  ( x  =  y  ->  D  =  [_ y  /  x ]_ D )
1210, 11ineq12d 3646 . . 3  |-  ( x  =  y  ->  ( C  i^i  D )  =  ( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D ) )
136, 9, 12csbief 3399 . 2  |-  [_ y  /  x ]_ ( C  i^i  D )  =  ( [_ y  /  x ]_ C  i^i  [_ y  /  x ]_ D )
145, 13vtoclg 3118 1  |-  ( A  e.  B  ->  [_ A  /  x ]_ ( C  i^i  D )  =  ( [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1454    e. wcel 1897   [_csb 3374    i^i cin 3414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-in 3422
This theorem is referenced by:  csbresgOLD  37255  onfrALTlem5VD  37321  onfrALTlem4VD  37322  csbresgVD  37331
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