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Theorem sbcimg 3032
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 2994 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] ( ph  ->  ps )  <->  [. A  /  x ]. ( ph  ->  ps ) ) )
2 dfsbcq2 2994 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
3 dfsbcq2 2994 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ps  <->  [. A  /  x ]. ps ) )
42, 3imbi12d 311 . 2  |-  ( y  =  A  ->  (
( [ y  /  x ] ph  ->  [ y  /  x ] ps ) 
<->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )
) )
5 sbim 2005 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
61, 4, 5vtoclbg 2844 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 176    = wceq 1623   [wsb 1629    e. wcel 1684   [.wsbc 2991
This theorem is referenced by:  sbceqal  3042  sbcimdv  3052  sbc19.21g  3055  sbcss  3564  tfinds2  4654  iota4an  5238  riotass2  6332  sbcfun  27985  sbcim2g  28302  sbcssOLD  28306  onfrALTlem5  28307  sbcim2gVD  28651  sbcssVD  28659  onfrALTlem5VD  28661  bnj538  28769  bnj110  28890  bnj92  28894  bnj539  28923  bnj540  28924  cdlemkid3N  31122  cdlemkid4  31123  cdlemk35s  31126  cdlemk39s  31128  cdlemk42  31130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992
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