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Theorem bnj985 29772
Description: Technical lemma for bnj69 29827. 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 29585 . . 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 29771 . . 3  |-  ( G  e.  _V  ->  ( G  e.  B  <->  [. G  / 
f ]. E. n ch ) )
62, 5ax-mp 5 . 2  |-  ( G  e.  B  <->  [. G  / 
f ]. E. n ch )
7 sbcex2 3350 . . 3  |-  ( [. G  /  f ]. E. p ch'  <->  E. p [. G  /  f ]. ch' )
8 nfv 1755 . . . . . . 7  |-  F/ p ch
98sb8e 2223 . . . . . 6  |-  ( E. n ch  <->  E. p [ p  /  n ] ch )
10 sbsbc 3303 . . . . . . 7  |-  ( [ p  /  n ] ch 
<-> 
[. p  /  n ]. ch )
1110exbii 1712 . . . . . 6  |-  ( E. p [ p  /  n ] ch  <->  E. p [. p  /  n ]. ch )
129, 11bitri 252 . . . . 5  |-  ( E. n ch  <->  E. p [. p  /  n ]. ch )
13 bnj985.6 . . . . 5  |-  ( ch'  <->  [. p  /  n ]. ch )
1412, 13bnj133 29541 . . . 4  |-  ( E. n ch  <->  E. p ch' )
1514sbcbii 3355 . . 3  |-  ( [. G  /  f ]. E. n ch  <->  [. G  /  f ]. E. p ch' )
16 bnj985.9 . . . 4  |-  ( ch"  <->  [. G  / 
f ]. ch' )
1716exbii 1712 . . 3  |-  ( E. p ch"  <->  E. p [. G  /  f ]. ch' )
187, 15, 173bitr4i 280 . 2  |-  ( [. G  /  f ]. E. n ch  <->  E. p ch" )
196, 18bitri 252 1  |-  ( G  e.  B  <->  E. p ch" )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ w3a 982    = wceq 1437   E.wex 1657   [wsb 1790    e. wcel 1872   {cab 2407   E.wrex 2772   _Vcvv 3080   [.wsbc 3299    u. cun 3434   {csn 3998   <.cop 4004    Fn wfn 5596    /\ w-bnj17 29499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-rex 2777  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-nul 3762  df-sn 3999  df-pr 4001  df-uni 4220  df-bnj17 29500
This theorem is referenced by:  bnj1018  29781
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