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Theorem csbresgOLD 37255
Description: Distribute proper substitution through the restriction of a class. csbresgOLD 37255 is derived from the virtual deduction proof csbresgVD 37331. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbres 5126 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
csbresgOLD  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B  |`  C )  =  (
[_ A  /  x ]_ B  |`  [_ A  /  x ]_ C ) )

Proof of Theorem csbresgOLD
StepHypRef Expression
1 csbingOLD 37254 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B  i^i  ( C  X.  _V ) )  =  (
[_ A  /  x ]_ B  i^i  [_ A  /  x ]_ ( C  X.  _V ) ) )
2 csbxpgOLD 37253 . . . . 5  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( C  X.  _V )  =  ( [_ A  /  x ]_ C  X.  [_ A  /  x ]_ _V ) )
3 csbconstg 3387 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ _V  =  _V )
43xpeq2d 4876 . . . . 5  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ C  X.  [_ A  /  x ]_ _V )  =  ( [_ A  /  x ]_ C  X.  _V ) )
52, 4eqtrd 2495 . . . 4  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( C  X.  _V )  =  ( [_ A  /  x ]_ C  X.  _V ) )
65ineq2d 3645 . . 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 2495 . 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 4864 . . 3  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
98csbeq2i 3793 . 2  |-  [_ A  /  x ]_ ( B  |`  C )  =  [_ A  /  x ]_ ( B  i^i  ( C  X.  _V ) )
10 df-res 4864 . 2  |-  ( [_ A  /  x ]_ B  |` 
[_ A  /  x ]_ C )  =  (
[_ A  /  x ]_ B  i^i  ( [_ A  /  x ]_ C  X.  _V )
)
117, 9, 103eqtr4g 2520 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B  |`  C )  =  (
[_ A  /  x ]_ B  |`  [_ A  /  x ]_ C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1454    e. wcel 1897   _Vcvv 3056   [_csb 3374    i^i cin 3414    X. cxp 4850    |` cres 4854
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  df-opab 4475  df-xp 4858  df-res 4864
This theorem is referenced by:  csbima12gALTOLD  37257  csbima12gALTVD  37333
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