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Theorem csbie2g 3140
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, 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 3095 . 2  |-  [_ A  /  x ]_ B  =  { z  |  [. A  /  x ]. z  e.  B }
2 csbie2g.1 . . . . 5  |-  ( x  =  y  ->  B  =  C )
32eleq2d 2363 . . . 4  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
4 csbie2g.2 . . . . 5  |-  ( y  =  A  ->  C  =  D )
54eleq2d 2363 . . . 4  |-  ( y  =  A  ->  (
z  e.  C  <->  z  e.  D ) )
63, 5sbcie2g 3037 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. z  e.  B  <->  z  e.  D ) )
76abbi1dv 2412 . 2  |-  ( A  e.  V  ->  { z  |  [. A  /  x ]. z  e.  B }  =  D )
81, 7syl5eq 2340 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {cab 2282   [.wsbc 3004   [_csb 3094
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  df-csb 3095
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