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Theorem bnj985 29301
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
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 29112 . . 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 29300 . . 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 3053 . . 3  |-  ( [. G  /  f ]. E. p ch'  <->  E. p [. G  /  f ]. ch' )
8 nfv 1609 . . . . . . 7  |-  F/ p ch
98sb8e 2046 . . . . . 6  |-  ( E. n ch  <->  E. p [ p  /  n ] ch )
10 sbsbc 3008 . . . . . . 7  |-  ( [ p  /  n ] ch 
<-> 
[. p  /  n ]. ch )
1110exbii 1572 . . . . . 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 29069 . . . 4  |-  ( E. n ch  <->  E. p ch' )
1514sbcbii 3059 . . 3  |-  ( [. G  /  f ]. E. n ch  <->  [. G  /  f ]. E. p ch' )
16 bnj985.9 . . . 4  |-  ( ch"  <->  [. G  / 
f ]. ch' )
1716exbii 1572 . . 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 1531    = wceq 1632   [wsb 1638    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801   [.wsbc 3004    u. cun 3163   {csn 3653   <.cop 3656    Fn wfn 5266    /\ w-bnj17 29027
This theorem is referenced by:  bnj1018  29310
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rex 2562  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-nul 3469  df-sn 3659  df-pr 3660  df-uni 3844  df-bnj17 29028
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