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Theorem bnj1441 29189
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 2565 . 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 1748 . . 3  |-  ( ( x  e.  A  /\  ph )  ->  A. y
( x  e.  A  /\  ph ) )
54hbab 2287 . 2  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  A. y  z  e.  {
x  |  ( x  e.  A  /\  ph ) } )
61, 5hbxfreq 2399 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 358   A.wal 1530    e. wcel 1696   {cab 2282   {crab 2560
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-rab 2565
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