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Theorem csbeq2d 2955
Description: Formula-building deduction for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
Hypotheses
Ref Expression
csbeq2d.1  |-  F/ x ph
csbeq2d.2  |-  ( ph  ->  B  =  C )
Assertion
Ref Expression
csbeq2d  |-  ( ph  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C )

Proof of Theorem csbeq2d
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq2d.1 . . . 4  |-  F/ x ph
2 csbeq2d.2 . . . . 5  |-  ( ph  ->  B  =  C )
32eleq2d 2157 . . . 4  |-  ( ph  ->  ( y  e.  B  <->  y  e.  C ) )
41, 3sbcbid 2896 . . 3  |-  ( ph  ->  ( [. A  /  x ]. y  e.  B  <->  [. A  /  x ]. y  e.  C )
)
54abbidv 2205 . 2  |-  ( ph  ->  { y  |  [. A  /  x ]. y  e.  B }  =  {
y  |  [. A  /  x ]. y  e.  C } )
6 df-csb 2934 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
7 df-csb 2934 . 2  |-  [_ A  /  x ]_ C  =  { y  |  [. A  /  x ]. y  e.  C }
85, 6, 73eqtr4g 2145 1  |-  ( ph  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289   F/wnf 1394    e. wcel 1438   {cab 2074   [.wsbc 2840   [_csb 2933
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-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-sbc 2841  df-csb 2934
This theorem is referenced by:  csbeq2dv  2956
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