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Theorem csbresg 4958
Description: Distribute proper substitution through the restriction of a class. csbresg 4958 is derived from the virtual deduction proof csbresgVD 28671. (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 3376 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B  i^i  ( C  X.  _V ) )  =  (
[_ A  /  x ]_ B  i^i  [_ A  /  x ]_ ( C  X.  _V ) ) )
2 csbxpg 4716 . . . . 5  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( C  X.  _V )  =  ( [_ A  /  x ]_ C  X.  [_ A  /  x ]_ _V ) )
3 csbconstg 3095 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ _V  =  _V )
43xpeq2d 4713 . . . . 5  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ C  X.  [_ A  /  x ]_ _V )  =  ( [_ A  /  x ]_ C  X.  _V ) )
52, 4eqtrd 2315 . . . 4  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( C  X.  _V )  =  ( [_ A  /  x ]_ C  X.  _V ) )
65ineq2d 3370 . . 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 2315 . 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 4701 . . 3  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
98csbeq2i 3107 . 2  |-  [_ A  /  x ]_ ( B  |`  C )  =  [_ A  /  x ]_ ( B  i^i  ( C  X.  _V ) )
10 df-res 4701 . 2  |-  ( [_ A  /  x ]_ B  |` 
[_ A  /  x ]_ C )  =  (
[_ A  /  x ]_ B  i^i  ( [_ A  /  x ]_ C  X.  _V )
)
117, 9, 103eqtr4g 2340 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 1623    e. wcel 1684   _Vcvv 2788   [_csb 3081    i^i cin 3151    X. cxp 4687    |` cres 4691
This theorem is referenced by:  csbima12gALT  5023  csbima12gALTVD  28673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-in 3159  df-opab 4078  df-xp 4695  df-res 4701
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