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Theorem bnj1133 29359
Description: Technical lemma for bnj69 29380. 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 29347 . . 3  |-  ( n  e.  D  ->  _E  Fr  n )
41, 3bnj769 29134 . 2  |-  ( ch 
->  _E  Fr  n )
5 impexp 444 . . . . . 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 1661 . . . 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 1643 . . 3  |-  A. i
( i  e.  n  ->  ( ta  ->  th )
)
10 df-ral 2758 . . 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 3061 . . 3  |-  n  e. 
_V
13 bnj1133.7 . . 3  |-  ( ta  <->  A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. th ) )
1412, 13bnj110 29230 . 2  |-  ( (  _E  Fr  n  /\  A. i  e.  n  ( ta  ->  th )
)  ->  A. i  e.  n  th )
154, 11, 14sylancl 660 1  |-  ( ch 
->  A. i  e.  n  th )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1403    = wceq 1405    e. wcel 1842   A.wral 2753   [.wsbc 3276    \ cdif 3410   (/)c0 3737   {csn 3971   class class class wbr 4394    _E cep 4731    Fr wfr 4778    Fn wfn 5563   omcom 6682    /\ w-bnj17 29052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-tr 4489  df-eprel 4733  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-om 6683  df-bnj17 29053
This theorem is referenced by:  bnj1128  29360
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