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Theorem sbcie2g 3289
Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3290 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 3257 . 2  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
2 sbcie2g.2 . 2  |-  ( y  =  A  ->  ( ps 
<->  ch ) )
3 sbsbc 3259 . . 3  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
4 nfv 1769 . . . 4  |-  F/ x ps
5 sbcie2g.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
64, 5sbie 2257 . . 3  |-  ( [ y  /  x ] ph 
<->  ps )
73, 6bitr3i 259 . 2  |-  ( [. y  /  x ]. ph  <->  ps )
81, 2, 7vtoclbg 3094 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1452   [wsb 1805    e. wcel 1904   [.wsbc 3255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-v 3033  df-sbc 3256
This theorem is referenced by:  sbcel2gv  3315  csbie2g  3380  brab1  4441  bnj90  29600  bnj124  29754  riotasvd  32592
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