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Theorem sbc19.21g 3227
Description: Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)
Hypothesis
Ref Expression
sbcgf.1  |-  F/ x ph
Assertion
Ref Expression
sbc19.21g  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  ->  ps ) 
<->  ( ph  ->  [. A  /  x ]. ps )
) )

Proof of Theorem sbc19.21g
StepHypRef Expression
1 sbcimg 3204 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  ->  ps ) 
<->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )
) )
2 sbcgf.1 . . . 4  |-  F/ x ph
32sbcgf 3226 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ph ) )
43imbi1d 310 . 2  |-  ( A  e.  V  ->  (
( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )  <->  ( ph  ->  [. A  /  x ]. ps ) ) )
51, 4bitrd 246 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  ->  ps ) 
<->  ( ph  ->  [. A  /  x ]. ps )
) )
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:  bnj121  29315  bnj124  29316  bnj130  29319  bnj207  29326  bnj611  29363  bnj1000  29386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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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