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Theorem sbcie2g 2914
Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 2915 avoids a disjointness condition on  x and  A by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
sbcie2g.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
sbcie2g.2  |-  ( y  =  A  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
sbcie2g  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ch ) )
Distinct variable groups:    x, y    y, A    ch, y    ph, y    ps, x
Allowed substitution hints:    ph( x)    ps( y)    ch( x)    A( x)    V( x, y)

Proof of Theorem sbcie2g
StepHypRef Expression
1 dfsbcq 2884 . 2  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
2 sbcie2g.2 . 2  |-  ( y  =  A  ->  ( ps 
<->  ch ) )
3 sbsbc 2886 . . 3  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
4 nfv 1493 . . . 4  |-  F/ x ps
5 sbcie2g.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
64, 5sbie 1749 . . 3  |-  ( [ y  /  x ] ph 
<->  ps )
73, 6bitr3i 185 . 2  |-  ( [. y  /  x ]. ph  <->  ps )
81, 2, 7vtoclbg 2721 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1316    e. wcel 1465   [wsb 1720   [.wsbc 2882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883
This theorem is referenced by:  sbcel2gv  2944  csbie2g  3020  brab1  3945
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