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Theorem sbcrel 4454
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 3358 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. R  C_  ( _V 
X.  _V )  <->  [_ A  /  x ]_ R  C_  [_ A  /  x ]_ ( _V 
X.  _V ) ) )
2 csbconstg 2892 . . . 4  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( _V 
X.  _V )  =  ( _V  X.  _V )
)
32sseq2d 3001 . . 3  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ R  C_  [_ A  /  x ]_ ( _V 
X.  _V )  <->  [_ A  /  x ]_ R  C_  ( _V  X.  _V ) ) )
41, 3bitrd 181 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. R  C_  ( _V 
X.  _V )  <->  [_ A  /  x ]_ R  C_  ( _V  X.  _V ) ) )
5 df-rel 4380 . . 3  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
65sbcbii 2845 . 2  |-  ( [. A  /  x ]. Rel  R  <->  [. A  /  x ]. R  C_  ( _V 
X.  _V ) )
7 df-rel 4380 . 2  |-  ( Rel  [_ A  /  x ]_ R  <->  [_ A  /  x ]_ R  C_  ( _V 
X.  _V ) )
84, 6, 73bitr4g 216 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. Rel  R  <->  Rel  [_ A  /  x ]_ R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102    e. wcel 1409   _Vcvv 2574   [.wsbc 2787   [_csb 2880    C_ wss 2945    X. cxp 4371   Rel wrel 4378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sbc 2788  df-csb 2881  df-in 2952  df-ss 2959  df-rel 4380
This theorem is referenced by:  sbcfung  4953
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