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Theorem sbcrel 27848
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 3698 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. R  C_  ( _V 
X.  _V )  <->  [_ A  /  x ]_ R  C_  [_ A  /  x ]_ ( _V 
X.  _V ) ) )
2 csbconstg 3225 . . . 4  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( _V 
X.  _V )  =  ( _V  X.  _V )
)
32sseq2d 3336 . . 3  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ R  C_  [_ A  /  x ]_ ( _V 
X.  _V )  <->  [_ A  /  x ]_ R  C_  ( _V  X.  _V ) ) )
41, 3bitrd 245 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. R  C_  ( _V 
X.  _V )  <->  [_ A  /  x ]_ R  C_  ( _V  X.  _V ) ) )
5 df-rel 4844 . . 3  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
65sbcbii 3176 . 2  |-  ( [. A  /  x ]. Rel  R  <->  [. A  /  x ]. R  C_  ( _V 
X.  _V ) )
7 df-rel 4844 . 2  |-  ( Rel  [_ A  /  x ]_ R  <->  [_ A  /  x ]_ R  C_  ( _V 
X.  _V ) )
84, 6, 73bitr4g 280 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. Rel  R  <->  Rel  [_ A  /  x ]_ R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1721   _Vcvv 2916   [.wsbc 3121   [_csb 3211    C_ wss 3280    X. cxp 4835   Rel wrel 4842
This theorem is referenced by:  sbcfun  27854
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-sbc 3122  df-csb 3212  df-in 3287  df-ss 3294  df-rel 4844
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