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Theorem sbcimg 2918
Description: Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)
Assertion
Ref Expression
sbcimg  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  ->  ps ) 
<->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )
) )

Proof of Theorem sbcimg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2881 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  ->  ps )  <->  [. A  /  x ]. ( ph  ->  ps ) ) )
2 dfsbcq2 2881 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
3 dfsbcq2 2881 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
42, 3imbi12d 233 . 2  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  ->  [ y  /  x ] ps ) 
<->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )
) )
5 sbim 1902 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
61, 4, 5vtoclbg 2718 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  ->  ps ) 
<->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1314    e. wcel 1463   [wsb 1718   [.wsbc 2878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-sbc 2879
This theorem is referenced by:  sbcim1  2925  sbceqal  2932  sbc19.21g  2945  sbcssg  3438  iota4an  5065  sbcfung  5105  riotass2  5710
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