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Theorem csbeq2 3030
Description: Substituting into equivalent classes gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
Assertion
Ref Expression
csbeq2  |-  ( A. x  B  =  C  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C )

Proof of Theorem csbeq2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eleq2 2204 . . . . 5  |-  ( B  =  C  ->  (
y  e.  B  <->  y  e.  C ) )
21alimi 1432 . . . 4  |-  ( A. x  B  =  C  ->  A. x ( y  e.  B  <->  y  e.  C ) )
3 sbcbi2 2962 . . . 4  |-  ( A. x ( y  e.  B  <->  y  e.  C
)  ->  ( [. A  /  x ]. y  e.  B  <->  [. A  /  x ]. y  e.  C
) )
42, 3syl 14 . . 3  |-  ( A. x  B  =  C  ->  ( [. A  /  x ]. y  e.  B  <->  [. A  /  x ]. y  e.  C )
)
54abbidv 2258 . 2  |-  ( A. x  B  =  C  ->  { y  |  [. A  /  x ]. y  e.  B }  =  {
y  |  [. A  /  x ]. y  e.  C } )
6 df-csb 3007 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
7 df-csb 3007 . 2  |-  [_ A  /  x ]_ C  =  { y  |  [. A  /  x ]. y  e.  C }
85, 6, 73eqtr4g 2198 1  |-  ( A. x  B  =  C  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1330    = wceq 1332    e. wcel 1481   {cab 2126   [.wsbc 2912   [_csb 3006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-sbc 2913  df-csb 3007
This theorem is referenced by:  prodeq2w  11356
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