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Theorem sbcgf 3226
Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Hypothesis
Ref Expression
sbcgf.1  |-  F/ x ph
Assertion
Ref Expression
sbcgf  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ph ) )

Proof of Theorem sbcgf
StepHypRef Expression
1 sbcgf.1 . 2  |-  F/ x ph
2 sbctt 3225 . 2  |-  ( ( A  e.  V  /\  F/ x ph )  -> 
( [. A  /  x ]. ph  <->  ph ) )
31, 2mpan2 654 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   F/wnf 1554    e. wcel 1726   [.wsbc 3163
This theorem is referenced by:  sbc19.21g  3227  sbcg  3228  sbcabel  3240  bnj110  29231  bnj1039  29342
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-sbc 3164
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