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Theorem sbcimg 3019
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 2980 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  ->  ps )  <->  [. A  /  x ]. ( ph  ->  ps ) ) )
2 dfsbcq2 2980 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
3 dfsbcq2 2980 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
42, 3imbi12d 234 . 2  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  ->  [ y  /  x ] ps ) 
<->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )
) )
5 sbim 1965 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
61, 4, 5vtoclbg 2813 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 105    = wceq 1364   [wsb 1773    e. wcel 2160   [.wsbc 2977
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978
This theorem is referenced by:  sbcim1  3026  sbceqal  3033  sbc19.21g  3046  sbcssg  3547  iota4an  5212  sbcfung  5255  riotass2  5873
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