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Theorem csbresg 4974
Description: Distribute proper substitution through the restriction of a class. csbresg 4974 is derived from the virtual deduction proof csbresgVD 28987. (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 3389 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B  i^i  ( C  X.  _V ) )  =  (
[_ A  /  x ]_ B  i^i  [_ A  /  x ]_ ( C  X.  _V ) ) )
2 csbxpg 4732 . . . . 5  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( C  X.  _V )  =  ( [_ A  /  x ]_ C  X.  [_ A  /  x ]_ _V ) )
3 csbconstg 3108 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ _V  =  _V )
43xpeq2d 4729 . . . . 5  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ C  X.  [_ A  /  x ]_ _V )  =  ( [_ A  /  x ]_ C  X.  _V ) )
52, 4eqtrd 2328 . . . 4  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( C  X.  _V )  =  ( [_ A  /  x ]_ C  X.  _V ) )
65ineq2d 3383 . . 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 2328 . 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 4717 . . 3  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
98csbeq2i 3120 . 2  |-  [_ A  /  x ]_ ( B  |`  C )  =  [_ A  /  x ]_ ( B  i^i  ( C  X.  _V ) )
10 df-res 4717 . 2  |-  ( [_ A  /  x ]_ B  |` 
[_ A  /  x ]_ C )  =  (
[_ A  /  x ]_ B  i^i  ( [_ A  /  x ]_ C  X.  _V )
)
117, 9, 103eqtr4g 2353 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 1632    e. wcel 1696   _Vcvv 2801   [_csb 3094    i^i cin 3164    X. cxp 4703    |` cres 4707
This theorem is referenced by:  csbima12gALT  5039  csbima12gALTVD  28989
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-in 3172  df-opab 4094  df-xp 4711  df-res 4717
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