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Theorem hbab 2161
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 2157 . 2  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
2 hbab.1 . . 3  |-  ( ph  ->  A. x ph )
32hbsb 1942 . 2  |-  ( [ z  /  y ]
ph  ->  A. x [ z  /  y ] ph )
41, 3hbxfrbi 1465 1  |-  ( z  e.  { y  | 
ph }  ->  A. x  z  e.  { y  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1346   [wsb 1755    e. wcel 2141   {cab 2156
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157
This theorem is referenced by:  nfsab  2162
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