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Theorem csbtt 2919
Description: Substitution doesn't affect a constant  B (in which  x is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
csbtt  |-  ( ( A  e.  V  /\  F/_ x B )  ->  [_ A  /  x ]_ B  =  B
)

Proof of Theorem csbtt
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-csb 2910 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 nfcr 2212 . . . 4  |-  ( F/_ x B  ->  F/ x  y  e.  B )
3 sbctt 2881 . . . 4  |-  ( ( A  e.  V  /\  F/ x  y  e.  B )  ->  ( [. A  /  x ]. y  e.  B  <->  y  e.  B ) )
42, 3sylan2 280 . . 3  |-  ( ( A  e.  V  /\  F/_ x B )  -> 
( [. A  /  x ]. y  e.  B  <->  y  e.  B ) )
54abbi1dv 2199 . 2  |-  ( ( A  e.  V  /\  F/_ x B )  ->  { y  |  [. A  /  x ]. y  e.  B }  =  B )
61, 5syl5eq 2126 1  |-  ( ( A  e.  V  /\  F/_ x B )  ->  [_ A  /  x ]_ B  =  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   F/wnf 1390    e. wcel 1434   {cab 2068   F/_wnfc 2207   [.wsbc 2816   [_csb 2909
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817  df-csb 2910
This theorem is referenced by:  csbconstgf  2920  sbnfc2  2963
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