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Theorem sbab 2578
Description: The right-hand side of the second equality is a way of representing proper substitution of  y for  x into a class variable. (Contributed by NM, 14-Sep-2003.)
Assertion
Ref Expression
sbab  |-  ( x  =  y  ->  A  =  { z  |  [
y  /  x ]
z  e.  A }
)
Distinct variable groups:    z, A    x, z    y, z
Allowed substitution hints:    A( x, y)

Proof of Theorem sbab
StepHypRef Expression
1 sbequ12 2083 . 2  |-  ( x  =  y  ->  (
z  e.  A  <->  [ y  /  x ] z  e.  A ) )
21abbi2dv 2570 1  |-  ( x  =  y  ->  A  =  { z  |  [
y  /  x ]
z  e.  A }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1444   [wsb 1797    e. wcel 1887   {cab 2437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447
This theorem is referenced by:  sbcel12  3772  sbceqg  3773  sbcel12gOLD  36905
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