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Theorem hbab1 2129
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
hbab1  |-  ( y  e.  { x  | 
ph }  ->  A. x  y  e.  { x  |  ph } )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem hbab1
StepHypRef Expression
1 df-clab 2127 . 2  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
2 hbs1 1912 . 2  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
31, 2hbxfrbi 1449 1  |-  ( y  e.  { x  | 
ph }  ->  A. x  y  e.  { x  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1330    e. wcel 1481   [wsb 1736   {cab 2126
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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515
This theorem depends on definitions:  df-bi 116  df-sb 1737  df-clab 2127
This theorem is referenced by:  nfsab1  2130  abeq2  2249
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