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Theorem hbab 2274
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 2270 . 2  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
2 hbab.1 . . 3  |-  ( ph  ->  A. x ph )
32hbsb 2049 . 2  |-  ( [ z  /  y ]
ph  ->  A. x [ z  /  y ] ph )
41, 3hbxfrbi 1555 1  |-  ( z  e.  { y  | 
ph }  ->  A. x  z  e.  { y  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527   [wsb 1629    e. wcel 1684   {cab 2269
This theorem is referenced by:  nfsab  2275  bnj1441  28873  bnj1309  29052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270
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