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Theorem sbc19.21g 3189
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 3166 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  ->  ps ) 
<->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )
) )
2 sbcgf.1 . . . 4  |-  F/ x ph
32sbcgf 3188 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ph ) )
43imbi1d 309 . 2  |-  ( A  e.  V  ->  (
( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )  <->  ( ph  ->  [. A  /  x ]. ps ) ) )
51, 4bitrd 245 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  ->  ps ) 
<->  ( ph  ->  [. A  /  x ]. ps )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   F/wnf 1550    e. wcel 1721   [.wsbc 3125
This theorem is referenced by:  bnj121  28951  bnj124  28952  bnj130  28955  bnj207  28962  bnj611  28999  bnj1000  29022
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-v 2922  df-sbc 3126
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