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Theorem bnj984 28662
Description: Technical lemma for bnj69 28718. 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
bnj984.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj984.11  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
Assertion
Ref Expression
bnj984  |-  ( G  e.  A  ->  ( G  e.  B  <->  [. G  / 
f ]. E. n ch ) )

Proof of Theorem bnj984
StepHypRef Expression
1 sbc8g 3112 . . 3  |-  ( G  e.  A  ->  ( [. G  /  f ]. E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )  <->  G  e.  { f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) } ) )
2 bnj984.11 . . . 4  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
32eleq2i 2452 . . 3  |-  ( G  e.  B  <->  G  e.  { f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) } )
41, 3syl6rbbr 256 . 2  |-  ( G  e.  A  ->  ( G  e.  B  <->  [. G  / 
f ]. E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) ) )
5 df-rex 2656 . . . 4  |-  ( E. n  e.  D  ( f  Fn  n  /\  ph 
/\  ps )  <->  E. n
( n  e.  D  /\  ( f  Fn  n  /\  ph  /\  ps )
) )
6 bnj984.3 . . . . 5  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
7 bnj252 28406 . . . . 5  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  <->  ( n  e.  D  /\  (
f  Fn  n  /\  ph 
/\  ps ) ) )
86, 7bitri 241 . . . 4  |-  ( ch  <->  ( n  e.  D  /\  ( f  Fn  n  /\  ph  /\  ps )
) )
95, 8bnj133 28431 . . 3  |-  ( E. n  e.  D  ( f  Fn  n  /\  ph 
/\  ps )  <->  E. n ch )
109sbcbiiOLD 3161 . 2  |-  ( G  e.  A  ->  ( [. G  /  f ]. E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )  <->  [. G  /  f ]. E. n ch ) )
114, 10bitrd 245 1  |-  ( G  e.  A  ->  ( G  e.  B  <->  [. G  / 
f ]. E. n ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2374   E.wrex 2651   [.wsbc 3105    Fn wfn 5390    /\ w-bnj17 28389
This theorem is referenced by:  bnj985  28663
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-rex 2656  df-v 2902  df-sbc 3106  df-bnj17 28390
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