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Theorem bnj1133 29360
Description: Technical lemma for bnj69 29381. 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 29348 . . 3  |-  ( n  e.  D  ->  _E  Fr  n )
41, 3bnj769 29133 . 2  |-  ( ch 
->  _E  Fr  n )
5 impexp 435 . . . . . 6  |-  ( ( ( i  e.  n  /\  ta )  ->  th )  <->  ( i  e.  n  -> 
( ta  ->  th )
) )
65bicomi 195 . . . . 5  |-  ( ( i  e.  n  -> 
( ta  ->  th )
)  <->  ( ( i  e.  n  /\  ta )  ->  th ) )
76albii 1576 . . . 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 1560 . . 3  |-  A. i
( i  e.  n  ->  ( ta  ->  th )
)
10 df-ral 2712 . . 3  |-  ( A. i  e.  n  ( ta  ->  th )  <->  A. i
( i  e.  n  ->  ( ta  ->  th )
) )
119, 10mpbir 202 . 2  |-  A. i  e.  n  ( ta  ->  th )
12 vex 2961 . . 3  |-  n  e. 
_V
13 bnj1133.7 . . 3  |-  ( ta  <->  A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. th ) )
1412, 13bnj110 29231 . 2  |-  ( (  _E  Fr  n  /\  A. i  e.  n  ( ta  ->  th )
)  ->  A. i  e.  n  th )
154, 11, 14sylancl 645 1  |-  ( ch 
->  A. i  e.  n  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726   A.wral 2707   [.wsbc 3163    \ cdif 3319   (/)c0 3630   {csn 3816   class class class wbr 4214    _E cep 4494    Fr wfr 4540   omcom 4847    Fn wfn 5451    /\ w-bnj17 29052
This theorem is referenced by:  bnj1128  29361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-bnj17 29053
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