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Theorem bnj1083 33113
Description: Technical lemma for bnj69 33145. 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 2820 . 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 2594 . 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 32835 . . . 4  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  <->  ( n  e.  D  /\  (
f  Fn  n  /\  ph 
/\  ps ) ) )
64, 5bitri 249 . . 3  |-  ( ch  <->  ( n  e.  D  /\  ( f  Fn  n  /\  ph  /\  ps )
) )
76exbii 1644 . 2  |-  ( E. n ch  <->  E. n
( n  e.  D  /\  ( f  Fn  n  /\  ph  /\  ps )
) )
81, 3, 73bitr4i 277 1  |-  ( f  e.  K  <->  E. n ch )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452   E.wrex 2815    Fn wfn 5581    /\ w-bnj17 32818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-12 1803  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-rex 2820  df-bnj17 32819
This theorem is referenced by:  bnj1121  33120  bnj1145  33128
  Copyright terms: Public domain W3C validator