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Theorem sbctt 2975
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 2912 . . . . 5  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
21bibi1d 232 . . . 4  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  <->  ph )  <->  ( [. A  /  x ]. ph  <->  ph ) ) )
32imbi2d 229 . . 3  |-  ( y  =  A  ->  (
( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )  <-> 
( F/ x ph  ->  ( [. A  /  x ]. ph  <->  ph ) ) ) )
4 sbft 1820 . . 3  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )
53, 4vtoclg 2746 . 2  |-  ( A  e.  V  ->  ( F/ x ph  ->  ( [. A  /  x ]. ph  <->  ph ) ) )
65imp 123 1  |-  ( ( A  e.  V  /\  F/ x ph )  -> 
( [. A  /  x ]. ph  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331   F/wnf 1436    e. wcel 1480   [wsb 1735   [.wsbc 2909
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910
This theorem is referenced by:  sbcgf  2976  csbtt  3014
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