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Theorem hbab 2106
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.)
Hypothesis
Ref Expression
hbab.1  |-  ( ph  ->  A. x ph )
Assertion
Ref Expression
hbab  |-  ( z  e.  { y  | 
ph }  ->  A. x  z  e.  { y  |  ph } )
Distinct variable group:    x, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem hbab
StepHypRef Expression
1 df-clab 2102 . 2  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
2 hbab.1 . . 3  |-  ( ph  ->  A. x ph )
32hbsb 1898 . 2  |-  ( [ z  /  y ]
ph  ->  A. x [ z  /  y ] ph )
41, 3hbxfrbi 1431 1  |-  ( z  e.  { y  | 
ph }  ->  A. x  z  e.  { y  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1312    e. wcel 1463   [wsb 1718   {cab 2101
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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102
This theorem is referenced by:  nfsab  2107
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