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Theorem sbcrel 28083
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 3577 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. R  C_  ( _V 
X.  _V )  <->  [_ A  /  x ]_ R  C_  [_ A  /  x ]_ ( _V 
X.  _V ) ) )
2 csbconstg 3108 . . . 4  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( _V 
X.  _V )  =  ( _V  X.  _V )
)
32sseq2d 3219 . . 3  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ R  C_  [_ A  /  x ]_ ( _V 
X.  _V )  <->  [_ A  /  x ]_ R  C_  ( _V  X.  _V ) ) )
41, 3bitrd 244 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. R  C_  ( _V 
X.  _V )  <->  [_ A  /  x ]_ R  C_  ( _V  X.  _V ) ) )
5 df-rel 4712 . . 3  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
65sbcbii 3059 . 2  |-  ( [. A  /  x ]. Rel  R  <->  [. A  /  x ]. R  C_  ( _V 
X.  _V ) )
7 df-rel 4712 . 2  |-  ( Rel  [_ A  /  x ]_ R  <->  [_ A  /  x ]_ R  C_  ( _V 
X.  _V ) )
84, 6, 73bitr4g 279 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. Rel  R  <->  Rel  [_ A  /  x ]_ R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1696   _Vcvv 2801   [.wsbc 3004   [_csb 3094    C_ wss 3165    X. cxp 4703   Rel wrel 4710
This theorem is referenced by:  sbcfun  28089
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005  df-csb 3095  df-in 3172  df-ss 3179  df-rel 4712
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