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Theorem sbcie2g 3186
Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3187 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 3155 . 2  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
2 sbcie2g.2 . 2  |-  ( y  =  A  ->  ( ps 
<->  ch ) )
3 sbsbc 3157 . . 3  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
4 nfv 1629 . . . 4  |-  F/ x ps
5 sbcie2g.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
64, 5sbie 2122 . . 3  |-  ( [ y  /  x ] ph 
<->  ps )
73, 6bitr3i 243 . 2  |-  ( [. y  /  x ]. ph  <->  ps )
81, 2, 7vtoclbg 3004 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652   [wsb 1658    e. wcel 1725   [.wsbc 3153
This theorem is referenced by:  sbcel2gv  3213  csbie2g  3289  brab1  4249  riotasvd  6584  bnj90  29024  bnj124  29179
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154
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