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Theorem sbctt 2824
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 2767 . . . . 5  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
21bibi1d 222 . . . 4  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  <->  ph )  <->  ( [. A  /  x ]. ph  <->  ph ) ) )
32imbi2d 219 . . 3  |-  ( y  =  A  ->  (
( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )  <-> 
( F/ x ph  ->  ( [. A  /  x ]. ph  <->  ph ) ) ) )
4 sbft 1728 . . 3  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )
53, 4vtoclg 2613 . 2  |-  ( A  e.  V  ->  ( F/ x ph  ->  ( [. A  /  x ]. ph  <->  ph ) ) )
65imp 115 1  |-  ( ( A  e.  V  /\  F/ x ph )  -> 
( [. A  /  x ]. ph  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243   F/wnf 1349    e. wcel 1393   [wsb 1645   [.wsbc 2764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765
This theorem is referenced by:  sbcgf  2825  csbtt  2862
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