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Theorem sbcrel 27971
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 sbcss 3740 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. R  C_  ( _V 
X.  _V )  <->  [_ A  /  x ]_ R  C_  [_ A  /  x ]_ ( _V 
X.  _V ) ) )
2 csbconstg 3267 . . . 4  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( _V 
X.  _V )  =  ( _V  X.  _V )
)
32sseq2d 3378 . . 3  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ R  C_  [_ A  /  x ]_ ( _V 
X.  _V )  <->  [_ A  /  x ]_ R  C_  ( _V  X.  _V ) ) )
41, 3bitrd 246 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. R  C_  ( _V 
X.  _V )  <->  [_ A  /  x ]_ R  C_  ( _V  X.  _V ) ) )
5 df-rel 4888 . . 3  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
65sbcbii 3218 . 2  |-  ( [. A  /  x ]. Rel  R  <->  [. A  /  x ]. R  C_  ( _V 
X.  _V ) )
7 df-rel 4888 . 2  |-  ( Rel  [_ A  /  x ]_ R  <->  [_ A  /  x ]_ R  C_  ( _V 
X.  _V ) )
84, 6, 73bitr4g 281 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. Rel  R  <->  Rel  [_ A  /  x ]_ R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    e. wcel 1726   _Vcvv 2958   [.wsbc 3163   [_csb 3253    C_ wss 3322    X. cxp 4879   Rel wrel 4886
This theorem is referenced by:  sbcfun  27977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-sbc 3164  df-csb 3254  df-in 3329  df-ss 3336  df-rel 4888
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