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Theorem bnj1133 29019
Description: Technical lemma for bnj69 29040. 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 29007 . . 3  |-  ( n  e.  D  ->  _E  Fr  n )
41, 3bnj769 28792 . 2  |-  ( ch 
->  _E  Fr  n )
5 impexp 433 . . . . . 6  |-  ( ( ( i  e.  n  /\  ta )  ->  th )  <->  ( i  e.  n  -> 
( ta  ->  th )
) )
65bicomi 193 . . . . 5  |-  ( ( i  e.  n  -> 
( ta  ->  th )
)  <->  ( ( i  e.  n  /\  ta )  ->  th ) )
76albii 1553 . . . 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 1537 . . 3  |-  A. i
( i  e.  n  ->  ( ta  ->  th )
)
10 df-ral 2548 . . 3  |-  ( A. i  e.  n  ( ta  ->  th )  <->  A. i
( i  e.  n  ->  ( ta  ->  th )
) )
119, 10mpbir 200 . 2  |-  A. i  e.  n  ( ta  ->  th )
12 vex 2791 . . 3  |-  n  e. 
_V
13 bnj1133.7 . . 3  |-  ( ta  <->  A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. th ) )
1412, 13bnj110 28890 . 2  |-  ( (  _E  Fr  n  /\  A. i  e.  n  ( ta  ->  th )
)  ->  A. i  e.  n  th )
154, 11, 14sylancl 643 1  |-  ( ch 
->  A. i  e.  n  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   A.wral 2543   [.wsbc 2991    \ cdif 3149   (/)c0 3455   {csn 3640   class class class wbr 4023    _E cep 4303    Fr wfr 4349   omcom 4656    Fn wfn 5250    /\ w-bnj17 28711
This theorem is referenced by:  bnj1128  29020
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-bnj17 28712
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