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Theorem bnj1133 32313
Description: Technical lemma for bnj69 32334. 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 32301 . . 3  |-  ( n  e.  D  ->  _E  Fr  n )
41, 3bnj769 32088 . 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 1611 . . . 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 1596 . . 3  |-  A. i
( i  e.  n  ->  ( ta  ->  th )
)
10 df-ral 2804 . . 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 3081 . . 3  |-  n  e. 
_V
13 bnj1133.7 . . 3  |-  ( ta  <->  A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. th ) )
1412, 13bnj110 32184 . 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 1368    = wceq 1370    e. wcel 1758   A.wral 2799   [.wsbc 3294    \ cdif 3434   (/)c0 3746   {csn 3986   class class class wbr 4401    _E cep 4739    Fr wfr 4785    Fn wfn 5522   omcom 6587    /\ w-bnj17 32007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-tr 4495  df-eprel 4741  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-om 6588  df-bnj17 32008
This theorem is referenced by:  bnj1128  32314
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