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Theorem hbab1 2376
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 2374 . 2  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
2 hbs1 2138 . 2  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
31, 2hbxfrbi 1574 1  |-  ( y  e.  { x  | 
ph }  ->  A. x  y  e.  { x  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1546   [wsb 1655    e. wcel 1717   {cab 2373
This theorem is referenced by:  nfsab1  2377  abeq2  2492  abeq2f  23804  bnj1317  28531  bnj1318  28732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374
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