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Theorem hbab1 2045
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 2043 . 2  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
2 hbs1 1830 . 2  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
31, 2hbxfrbi 1377 1  |-  ( y  e.  { x  | 
ph }  ->  A. x  y  e.  { x  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1257    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-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
This theorem depends on definitions:  df-bi 114  df-sb 1662  df-clab 2043
This theorem is referenced by:  nfsab1  2046  abeq2  2162
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