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Theorem sbcimg 3203
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 3165 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  ->  ps )  <->  [. A  /  x ]. ( ph  ->  ps ) ) )
2 dfsbcq2 3165 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
3 dfsbcq2 3165 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
42, 3imbi12d 313 . 2  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  ->  [ y  /  x ] ps ) 
<->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )
) )
5 sbim 2136 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
61, 4, 5vtoclbg 3013 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 178    = wceq 1653   [wsb 1659    e. wcel 1726   [.wsbc 3162
This theorem is referenced by:  sbceqal  3213  sbcimdv  3223  sbc19.21g  3226  sbcss  3739  tfinds2  4844  iota4an  5438  riotass2  6578  sbcfun  27964  sbcim2g  28624  sbcssOLD  28628  onfrALTlem5  28629  sbcim2gVD  28988  sbcssVD  28996  onfrALTlem5VD  28998  bnj538  29109  bnj110  29230  bnj92  29234  bnj539  29263  bnj540  29264  cdlemkid3N  31731  cdlemkid4  31732  cdlemk35s  31735  cdlemk39s  31737  cdlemk42  31739
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2418
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 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959  df-sbc 3163
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