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Theorem sbctt 2890
Description: Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
sbctt  |-  ( ( A  e.  V  /\  F/ x ph )  -> 
( [. A  /  x ]. ph  <->  ph ) )

Proof of Theorem sbctt
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2828 . . . . 5  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
21bibi1d 231 . . . 4  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  <->  ph )  <->  ( [. A  /  x ]. ph  <->  ph ) ) )
32imbi2d 228 . . 3  |-  ( y  =  A  ->  (
( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )  <-> 
( F/ x ph  ->  ( [. A  /  x ]. ph  <->  ph ) ) ) )
4 sbft 1771 . . 3  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )
53, 4vtoclg 2667 . 2  |-  ( A  e.  V  ->  ( F/ x ph  ->  ( [. A  /  x ]. ph  <->  ph ) ) )
65imp 122 1  |-  ( ( A  e.  V  /\  F/ x ph )  -> 
( [. A  /  x ]. ph  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   F/wnf 1390    e. wcel 1434   [wsb 1687   [.wsbc 2825
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 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-sbc 2826
This theorem is referenced by:  sbcgf  2891  csbtt  2928
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