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Theorem csbresg 5108
Description: Distribute proper substitution through the restriction of a class. csbresg 5108 is derived from the virtual deduction proof csbresgVD 28716. (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 3508 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B  i^i  ( C  X.  _V ) )  =  (
[_ A  /  x ]_ B  i^i  [_ A  /  x ]_ ( C  X.  _V ) ) )
2 csbxpg 4864 . . . . 5  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( C  X.  _V )  =  ( [_ A  /  x ]_ C  X.  [_ A  /  x ]_ _V ) )
3 csbconstg 3225 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ _V  =  _V )
43xpeq2d 4861 . . . . 5  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ C  X.  [_ A  /  x ]_ _V )  =  ( [_ A  /  x ]_ C  X.  _V ) )
52, 4eqtrd 2436 . . . 4  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( C  X.  _V )  =  ( [_ A  /  x ]_ C  X.  _V ) )
65ineq2d 3502 . . 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 2436 . 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 4849 . . 3  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
98csbeq2i 3237 . 2  |-  [_ A  /  x ]_ ( B  |`  C )  =  [_ A  /  x ]_ ( B  i^i  ( C  X.  _V ) )
10 df-res 4849 . 2  |-  ( [_ A  /  x ]_ B  |` 
[_ A  /  x ]_ C )  =  (
[_ A  /  x ]_ B  i^i  ( [_ A  /  x ]_ C  X.  _V )
)
117, 9, 103eqtr4g 2461 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 1649    e. wcel 1721   _Vcvv 2916   [_csb 3211    i^i cin 3279    X. cxp 4835    |` cres 4839
This theorem is referenced by:  csbima12gALT  5173  csbima12gALTVD  28718
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-in 3287  df-opab 4227  df-xp 4843  df-res 4849
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