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Theorem hbab1 2285
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 2283 . 2  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
2 hbs1 2057 . 2  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
31, 2hbxfrbi 1558 1  |-  ( y  e.  { x  | 
ph }  ->  A. x  y  e.  { x  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530   [wsb 1638    e. wcel 1696   {cab 2282
This theorem is referenced by:  nfsab1  2286  abeq2  2401  abeq2f  23145  rabid2f  23151  bnj1317  29170  bnj1318  29371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283
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