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Theorem sbcie2g 2856
Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 2857 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 2826 . 2  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
2 sbcie2g.2 . 2  |-  ( y  =  A  ->  ( ps 
<->  ch ) )
3 sbsbc 2828 . . 3  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
4 nfv 1462 . . . 4  |-  F/ x ps
5 sbcie2g.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
64, 5sbie 1716 . . 3  |-  ( [ y  /  x ] ph 
<->  ps )
73, 6bitr3i 184 . 2  |-  ( [. y  /  x ]. ph  <->  ps )
81, 2, 7vtoclbg 2668 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285    e. wcel 1434   [wsb 1687   [.wsbc 2824
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-sbc 2825
This theorem is referenced by:  sbcel2gv  2886  csbie2g  2961  brab1  3850
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