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Theorem sbc19.21g 3031
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 3004 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  ->  ps ) 
<->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )
) )
2 sbcgf.1 . . . 4  |-  F/ x ph
32sbcgf 3030 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ph ) )
43imbi1d 231 . 2  |-  ( A  e.  V  ->  (
( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )  <->  ( ph  ->  [. A  /  x ]. ps ) ) )
51, 4bitrd 188 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  ->  ps ) 
<->  ( ph  ->  [. A  /  x ]. ps )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   F/wnf 1460    e. wcel 2148   [.wsbc 2962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963
This theorem is referenced by: (None)
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