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Theorem bnj1083 29324
Description: Technical lemma for bnj69 29356. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1083.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj1083.8  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
Assertion
Ref Expression
bnj1083  |-  ( f  e.  K  <->  E. n ch )

Proof of Theorem bnj1083
StepHypRef Expression
1 df-rex 2562 . 2  |-  ( E. n  e.  D  ( f  Fn  n  /\  ph 
/\  ps )  <->  E. n
( n  e.  D  /\  ( f  Fn  n  /\  ph  /\  ps )
) )
2 bnj1083.8 . . 3  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
32abeq2i 2403 . 2  |-  ( f  e.  K  <->  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) )
4 bnj1083.3 . . . 4  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
5 bnj252 29044 . . . 4  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  <->  ( n  e.  D  /\  (
f  Fn  n  /\  ph 
/\  ps ) ) )
64, 5bitri 240 . . 3  |-  ( ch  <->  ( n  e.  D  /\  ( f  Fn  n  /\  ph  /\  ps )
) )
76exbii 1572 . 2  |-  ( E. n ch  <->  E. n
( n  e.  D  /\  ( f  Fn  n  /\  ph  /\  ps )
) )
81, 3, 73bitr4i 268 1  |-  ( f  e.  K  <->  E. n ch )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557    Fn wfn 5266    /\ w-bnj17 29027
This theorem is referenced by:  bnj1121  29331  bnj1145  29339
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-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1532  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-rex 2562  df-bnj17 29028
  Copyright terms: Public domain W3C validator