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Theorem sbab 2418
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 1872 . 2  |-  ( x  =  y  ->  (
z  e.  A  <->  [ y  /  x ] z  e.  A ) )
21abbi2dv 2411 1  |-  ( x  =  y  ->  A  =  { z  |  [
y  /  x ]
z  e.  A }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   [wsb 1638    e. wcel 1696   {cab 2282
This theorem is referenced by:  sbcel12g  3109  sbceqg  3110
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-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292
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