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Theorem sbcie2g 3037
Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3038 avoids a disjointness condition on  x ,  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 3006 . 2  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
2 sbcie2g.2 . 2  |-  ( y  =  A  ->  ( ps 
<->  ch ) )
3 sbsbc 3008 . . 3  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
4 nfv 1609 . . . 4  |-  F/ x ps
5 sbcie2g.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
64, 5sbie 1991 . . 3  |-  ( [ y  /  x ] ph 
<->  ps )
73, 6bitr3i 242 . 2  |-  ( [. y  /  x ]. ph  <->  ps )
81, 2, 7vtoclbg 2857 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632   [wsb 1638    e. wcel 1696   [.wsbc 3004
This theorem is referenced by:  csbie2g  3140  brab1  4084  riotasvd  6363  bnj90  29064  bnj124  29219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005
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