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Theorem bnj1441 29149
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1441.1  |-  ( x  e.  A  ->  A. y  x  e.  A )
bnj1441.2  |-  ( ph  ->  A. y ph )
Assertion
Ref Expression
bnj1441  |-  ( z  e.  { x  e.  A  |  ph }  ->  A. y  z  e. 
{ x  e.  A  |  ph } )
Distinct variable group:    y, z
Allowed substitution hints:    ph( x, y, z)    A( x, y, z)

Proof of Theorem bnj1441
StepHypRef Expression
1 df-rab 2706 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 bnj1441.1 . . . 4  |-  ( x  e.  A  ->  A. y  x  e.  A )
3 bnj1441.2 . . . 4  |-  ( ph  ->  A. y ph )
42, 3hban 1850 . . 3  |-  ( ( x  e.  A  /\  ph )  ->  A. y
( x  e.  A  /\  ph ) )
54hbab 2426 . 2  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  A. y  z  e.  {
x  |  ( x  e.  A  /\  ph ) } )
61, 5hbxfreq 2538 1  |-  ( z  e.  { x  e.  A  |  ph }  ->  A. y  z  e. 
{ x  e.  A  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1549    e. wcel 1725   {cab 2421   {crab 2701
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-rab 2706
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