ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  csbtt Unicode version

Theorem csbtt 2983
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 2974 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 nfcr 2248 . . . 4  |-  ( F/_ x B  ->  F/ x  y  e.  B )
3 sbctt 2945 . . . 4  |-  ( ( A  e.  V  /\  F/ x  y  e.  B )  ->  ( [. A  /  x ]. y  e.  B  <->  y  e.  B ) )
42, 3sylan2 282 . . 3  |-  ( ( A  e.  V  /\  F/_ x B )  -> 
( [. A  /  x ]. y  e.  B  <->  y  e.  B ) )
54abbi1dv 2235 . 2  |-  ( ( A  e.  V  /\  F/_ x B )  ->  { y  |  [. A  /  x ]. y  e.  B }  =  B )
61, 5syl5eq 2160 1  |-  ( ( A  e.  V  /\  F/_ x B )  ->  [_ A  /  x ]_ B  =  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314   F/wnf 1419    e. wcel 1463   {cab 2101   F/_wnfc 2243   [.wsbc 2880   [_csb 2973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-sbc 2881  df-csb 2974
This theorem is referenced by:  csbconstgf  2984  sbnfc2  3028
  Copyright terms: Public domain W3C validator