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Theorem bnj984 28984
Description: Technical lemma for bnj69 29040. 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 df-sbc 2992 . . . 4  |-  ( [. G  /  f ]. E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps )  <->  G  e.  { f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) } )
21a1i 10 . . 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 ) } ) )
3 bnj984.11 . . . 4  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
43eleq2i 2347 . . 3  |-  ( G  e.  B  <->  G  e.  { f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) } )
52, 4syl6rbbr 255 . 2  |-  ( G  e.  A  ->  ( G  e.  B  <->  [. G  / 
f ]. E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) ) )
6 df-rex 2549 . . . 4  |-  ( E. n  e.  D  ( f  Fn  n  /\  ph 
/\  ps )  <->  E. n
( n  e.  D  /\  ( f  Fn  n  /\  ph  /\  ps )
) )
7 bnj984.3 . . . . 5  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
8 bnj252 28728 . . . . 5  |-  ( ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps )  <->  ( n  e.  D  /\  (
f  Fn  n  /\  ph 
/\  ps ) ) )
97, 8bitri 240 . . . 4  |-  ( ch  <->  ( n  e.  D  /\  ( f  Fn  n  /\  ph  /\  ps )
) )
106, 9bnj133 28753 . . 3  |-  ( E. n  e.  D  ( f  Fn  n  /\  ph 
/\  ps )  <->  E. n ch )
1110sbcbiiOLD 3047 . 2  |-  ( G  e.  A  ->  ( [. G  /  f ]. E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )  <->  [. G  /  f ]. E. n ch ) )
125, 11bitrd 244 1  |-  ( G  e.  A  ->  ( G  e.  B  <->  [. G  / 
f ]. E. n ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   [.wsbc 2991    Fn wfn 5250    /\ w-bnj17 28711
This theorem is referenced by:  bnj985  28985
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-rex 2549  df-sbc 2992  df-bnj17 28712
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