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Theorem sbcrel 4512
Description: Distribute proper substitution through a relation predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
Assertion
Ref Expression
sbcrel  |-  ( A  e.  V  ->  ( [. A  /  x ]. Rel  R  <->  Rel  [_ A  /  x ]_ R ) )

Proof of Theorem sbcrel
StepHypRef Expression
1 sbcssg 3387 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. R  C_  ( _V 
X.  _V )  <->  [_ A  /  x ]_ R  C_  [_ A  /  x ]_ ( _V 
X.  _V ) ) )
2 csbconstg 2943 . . . 4  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( _V 
X.  _V )  =  ( _V  X.  _V )
)
32sseq2d 3052 . . 3  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ R  C_  [_ A  /  x ]_ ( _V 
X.  _V )  <->  [_ A  /  x ]_ R  C_  ( _V  X.  _V ) ) )
41, 3bitrd 186 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. R  C_  ( _V 
X.  _V )  <->  [_ A  /  x ]_ R  C_  ( _V  X.  _V ) ) )
5 df-rel 4435 . . 3  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
65sbcbii 2896 . 2  |-  ( [. A  /  x ]. Rel  R  <->  [. A  /  x ]. R  C_  ( _V 
X.  _V ) )
7 df-rel 4435 . 2  |-  ( Rel  [_ A  /  x ]_ R  <->  [_ A  /  x ]_ R  C_  ( _V 
X.  _V ) )
84, 6, 73bitr4g 221 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. Rel  R  <->  Rel  [_ A  /  x ]_ R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    e. wcel 1438   _Vcvv 2619   [.wsbc 2838   [_csb 2931    C_ wss 2997    X. cxp 4426   Rel wrel 4433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2839  df-csb 2932  df-in 3003  df-ss 3010  df-rel 4435
This theorem is referenced by:  sbcfung  5025
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