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Theorem sbab 2267
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 1744 . 2  |-  ( x  =  y  ->  (
z  e.  A  <->  [ y  /  x ] z  e.  A ) )
21abbi2dv 2258 1  |-  ( x  =  y  ->  A  =  { z  |  [
y  /  x ]
z  e.  A }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480   [wsb 1735   {cab 2125
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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135
This theorem is referenced by:  sbcel12g  3017  sbceqg  3018
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