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Theorem sbctt 2905
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 2843 . . . . 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 1776 . . 3  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )
53, 4vtoclg 2679 . 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 1289   F/wnf 1394    e. wcel 1438   [wsb 1692   [.wsbc 2840
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841
This theorem is referenced by:  sbcgf  2906  csbtt  2943
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