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Theorem sbab 2180
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 1670 . 2  |-  ( x  =  y  ->  (
z  e.  A  <->  [ y  /  x ] z  e.  A ) )
21abbi2dv 2172 1  |-  ( x  =  y  ->  A  =  { z  |  [
y  /  x ]
z  e.  A }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1259    e. wcel 1409   [wsb 1661   {cab 2042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052
This theorem is referenced by:  sbcel12g  2893  sbceqg  2894
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