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Theorem bnj985 28985
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
bnj985.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj985.6  |-  ( ch'  <->  [. p  /  n ]. ch )
bnj985.9  |-  ( ch"  <->  [. G  / 
f ]. ch' )
bnj985.11  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
bnj985.13  |-  G  =  ( f  u.  { <. n ,  C >. } )
Assertion
Ref Expression
bnj985  |-  ( G  e.  B  <->  E. p ch" )
Distinct variable groups:    G, p    ch, p    f, p
Allowed substitution hints:    ph( f, n, p)    ps( f, n, p)    ch( f, n)    B( f, n, p)    C( f, n, p)    D( f, n, p)    G( f, n)    ch'( f, n, p)   
ch"( f, n, p)

Proof of Theorem bnj985
StepHypRef Expression
1 bnj985.13 . . . 4  |-  G  =  ( f  u.  { <. n ,  C >. } )
21bnj918 28796 . . 3  |-  G  e. 
_V
3 bnj985.3 . . . 4  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
4 bnj985.11 . . . 4  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
53, 4bnj984 28984 . . 3  |-  ( G  e.  _V  ->  ( G  e.  B  <->  [. G  / 
f ]. E. n ch ) )
62, 5ax-mp 8 . 2  |-  ( G  e.  B  <->  [. G  / 
f ]. E. n ch )
7 sbcex2 3040 . . 3  |-  ( [. G  /  f ]. E. p ch'  <->  E. p [. G  /  f ]. ch' )
8 nfv 1605 . . . . . . 7  |-  F/ p ch
98sb8e 2033 . . . . . 6  |-  ( E. n ch  <->  E. p [ p  /  n ] ch )
10 sbsbc 2995 . . . . . . 7  |-  ( [ p  /  n ] ch 
<-> 
[. p  /  n ]. ch )
1110exbii 1569 . . . . . 6  |-  ( E. p [ p  /  n ] ch  <->  E. p [. p  /  n ]. ch )
129, 11bitri 240 . . . . 5  |-  ( E. n ch  <->  E. p [. p  /  n ]. ch )
13 bnj985.6 . . . . 5  |-  ( ch'  <->  [. p  /  n ]. ch )
1412, 13bnj133 28753 . . . 4  |-  ( E. n ch  <->  E. p ch' )
1514sbcbii 3046 . . 3  |-  ( [. G  /  f ]. E. n ch  <->  [. G  /  f ]. E. p ch' )
16 bnj985.9 . . . 4  |-  ( ch"  <->  [. G  / 
f ]. ch' )
1716exbii 1569 . . 3  |-  ( E. p ch"  <->  E. p [. G  /  f ]. ch' )
187, 15, 173bitr4i 268 . 2  |-  ( [. G  /  f ]. E. n ch  <->  E. p ch" )
196, 18bitri 240 1  |-  ( G  e.  B  <->  E. p ch" )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ w3a 934   E.wex 1528    = wceq 1623   [wsb 1629    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788   [.wsbc 2991    u. cun 3150   {csn 3640   <.cop 3643    Fn wfn 5250    /\ w-bnj17 28711
This theorem is referenced by:  bnj1018  28994
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  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-nfc 2408  df-ne 2448  df-rex 2549  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-nul 3456  df-sn 3646  df-pr 3647  df-uni 3828  df-bnj17 28712
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