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Theorem bnj1133 33124
Description: Technical lemma for bnj69 33145. 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
bnj1133.3  |-  D  =  ( om  \  { (/)
} )
bnj1133.5  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj1133.7  |-  ( ta  <->  A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. th ) )
bnj1133.8  |-  ( ( i  e.  n  /\  ta )  ->  th )
Assertion
Ref Expression
bnj1133  |-  ( ch 
->  A. i  e.  n  th )
Distinct variable groups:    i, j, n    th, j
Allowed substitution hints:    ph( f, i, j, n)    ps( f,
i, j, n)    ch( f, i, j, n)    th( f,
i, n)    ta( f,
i, j, n)    D( f, i, j, n)

Proof of Theorem bnj1133
StepHypRef Expression
1 bnj1133.5 . . 3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
2 bnj1133.3 . . . 4  |-  D  =  ( om  \  { (/)
} )
32bnj1071 33112 . . 3  |-  ( n  e.  D  ->  _E  Fr  n )
41, 3bnj769 32899 . 2  |-  ( ch 
->  _E  Fr  n )
5 impexp 446 . . . . . 6  |-  ( ( ( i  e.  n  /\  ta )  ->  th )  <->  ( i  e.  n  -> 
( ta  ->  th )
) )
65bicomi 202 . . . . 5  |-  ( ( i  e.  n  -> 
( ta  ->  th )
)  <->  ( ( i  e.  n  /\  ta )  ->  th ) )
76albii 1620 . . . 4  |-  ( A. i ( i  e.  n  ->  ( ta  ->  th ) )  <->  A. i
( ( i  e.  n  /\  ta )  ->  th ) )
8 bnj1133.8 . . . 4  |-  ( ( i  e.  n  /\  ta )  ->  th )
97, 8mpgbir 1605 . . 3  |-  A. i
( i  e.  n  ->  ( ta  ->  th )
)
10 df-ral 2819 . . 3  |-  ( A. i  e.  n  ( ta  ->  th )  <->  A. i
( i  e.  n  ->  ( ta  ->  th )
) )
119, 10mpbir 209 . 2  |-  A. i  e.  n  ( ta  ->  th )
12 vex 3116 . . 3  |-  n  e. 
_V
13 bnj1133.7 . . 3  |-  ( ta  <->  A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. th ) )
1412, 13bnj110 32995 . 2  |-  ( (  _E  Fr  n  /\  A. i  e.  n  ( ta  ->  th )
)  ->  A. i  e.  n  th )
154, 11, 14sylancl 662 1  |-  ( ch 
->  A. i  e.  n  th )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1377    = wceq 1379    e. wcel 1767   A.wral 2814   [.wsbc 3331    \ cdif 3473   (/)c0 3785   {csn 4027   class class class wbr 4447    _E cep 4789    Fr wfr 4835    Fn wfn 5581   omcom 6678    /\ w-bnj17 32818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-om 6679  df-bnj17 32819
This theorem is referenced by:  bnj1128  33125
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