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Theorem csbie2g 3240
Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3138 avoids a disjointness condition on  x ,  A by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)
Hypotheses
Ref Expression
csbie2g.1  |-  ( x  =  y  ->  B  =  C )
csbie2g.2  |-  ( y  =  A  ->  C  =  D )
Assertion
Ref Expression
csbie2g  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  D )
Distinct variable groups:    x, y    y, A    y, B    x, C    y, D
Allowed substitution hints:    A( x)    B( x)    C( y)    D( x)    V( x, y)

Proof of Theorem csbie2g
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-csb 3195 . 2  |-  [_ A  /  x ]_ B  =  { z  |  [. A  /  x ]. z  e.  B }
2 csbie2g.1 . . . . 5  |-  ( x  =  y  ->  B  =  C )
32eleq2d 2454 . . . 4  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
4 csbie2g.2 . . . . 5  |-  ( y  =  A  ->  C  =  D )
54eleq2d 2454 . . . 4  |-  ( y  =  A  ->  (
z  e.  C  <->  z  e.  D ) )
63, 5sbcie2g 3137 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. z  e.  B  <->  z  e.  D ) )
76abbi1dv 2503 . 2  |-  ( A  e.  V  ->  { z  |  [. A  /  x ]. z  e.  B }  =  D )
81, 7syl5eq 2431 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   {cab 2373   [.wsbc 3104   [_csb 3194
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-sbc 3105  df-csb 3195
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