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Theorem hbab 2148
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 2144 . 2  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
2 hbab.1 . . 3  |-  ( ph  ->  A. x ph )
32hbsb 1929 . 2  |-  ( [ z  /  y ]
ph  ->  A. x [ z  /  y ] ph )
41, 3hbxfrbi 1452 1  |-  ( z  e.  { y  | 
ph }  ->  A. x  z  e.  { y  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1333   [wsb 1742    e. wcel 2128   {cab 2143
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144
This theorem is referenced by:  nfsab  2149
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