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Theorem sbcgf 3067
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 3066 . 2  |-  ( ( A  e.  V  /\  F/ x ph )  -> 
( [. A  /  x ]. ph  <->  ph ) )
31, 2mpan2 652 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   F/wnf 1534    e. wcel 1696   [.wsbc 3004
This theorem is referenced by:  sbc19.21g  3068  sbcg  3069  sbcabel  3081  bnj110  29206  bnj1039  29317  bnj1128  29336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005
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