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Theorem hbab1 2370
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 26-May-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 2368 . 2  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
2 hbs1 2184 . 2  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
31, 2hbxfrbi 1651 1  |-  ( y  e.  { x  | 
ph }  ->  A. x  y  e.  { x  |  ph } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1397   [wsb 1747    e. wcel 1826   {cab 2367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-12 1862  ax-13 2006
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368
This theorem is referenced by:  nfsab1  2371  abeq2  2506  abbi  2513  abeq2f  27515  bnj1317  34227  bnj1318  34428
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