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Theorem csbie2g 2963
Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 2859 avoids a disjointness condition on  x and  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 2920 . 2  |-  [_ A  /  x ]_ B  =  { z  |  [. A  /  x ]. z  e.  B }
2 csbie2g.1 . . . . 5  |-  ( x  =  y  ->  B  =  C )
32eleq2d 2152 . . . 4  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
4 csbie2g.2 . . . . 5  |-  ( y  =  A  ->  C  =  D )
54eleq2d 2152 . . . 4  |-  ( y  =  A  ->  (
z  e.  C  <->  z  e.  D ) )
63, 5sbcie2g 2858 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. z  e.  B  <->  z  e.  D ) )
76abbi1dv 2202 . 2  |-  ( A  e.  V  ->  { z  |  [. A  /  x ]. z  e.  B }  =  D )
81, 7syl5eq 2127 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   {cab 2069   [.wsbc 2826   [_csb 2919
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 2614  df-sbc 2827  df-csb 2920
This theorem is referenced by: (None)
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