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Theorem csbresg 5141
Description: Distribute proper substitution through the restriction of a class. csbresg 5141 is derived from the virtual deduction proof csbresgVD 28934. (Contributed by Alan Sare, 10-Nov-2012.)
Assertion
Ref Expression
csbresg  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B  |`  C )  =  (
[_ A  /  x ]_ B  |`  [_ A  /  x ]_ C ) )

Proof of Theorem csbresg
StepHypRef Expression
1 csbing 3540 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B  i^i  ( C  X.  _V ) )  =  (
[_ A  /  x ]_ B  i^i  [_ A  /  x ]_ ( C  X.  _V ) ) )
2 csbxpg 4897 . . . . 5  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( C  X.  _V )  =  ( [_ A  /  x ]_ C  X.  [_ A  /  x ]_ _V ) )
3 csbconstg 3257 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ _V  =  _V )
43xpeq2d 4894 . . . . 5  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ C  X.  [_ A  /  x ]_ _V )  =  ( [_ A  /  x ]_ C  X.  _V ) )
52, 4eqtrd 2467 . . . 4  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( C  X.  _V )  =  ( [_ A  /  x ]_ C  X.  _V ) )
65ineq2d 3534 . . 3  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ B  i^i  [_ A  /  x ]_ ( C  X.  _V ) )  =  ( [_ A  /  x ]_ B  i^i  ( [_ A  /  x ]_ C  X.  _V )
) )
71, 6eqtrd 2467 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B  i^i  ( C  X.  _V ) )  =  (
[_ A  /  x ]_ B  i^i  ( [_ A  /  x ]_ C  X.  _V )
) )
8 df-res 4882 . . 3  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
98csbeq2i 3269 . 2  |-  [_ A  /  x ]_ ( B  |`  C )  =  [_ A  /  x ]_ ( B  i^i  ( C  X.  _V ) )
10 df-res 4882 . 2  |-  ( [_ A  /  x ]_ B  |` 
[_ A  /  x ]_ C )  =  (
[_ A  /  x ]_ B  i^i  ( [_ A  /  x ]_ C  X.  _V )
)
117, 9, 103eqtr4g 2492 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B  |`  C )  =  (
[_ A  /  x ]_ B  |`  [_ A  /  x ]_ C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948   [_csb 3243    i^i cin 3311    X. cxp 4868    |` cres 4872
This theorem is referenced by:  csbima12gALT  5206  csbima12gALTVD  28936
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  df-opab 4259  df-xp 4876  df-res 4882
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