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Theorem bnj986 28986
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
bnj986.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj986.10  |-  D  =  ( om  \  { (/)
} )
bnj986.15  |-  ( ta  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )
Assertion
Ref Expression
bnj986  |-  ( ch 
->  E. m E. p ta )
Distinct variable group:    m, n, p
Allowed substitution hints:    ph( f, m, n, p)    ps( f, m, n, p)    ch( f, m, n, p)    ta( f, m, n, p)    D( f, m, n, p)

Proof of Theorem bnj986
StepHypRef Expression
1 bnj986.3 . . . . . 6  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
2 bnj986.10 . . . . . . 7  |-  D  =  ( om  \  { (/)
} )
32bnj158 28757 . . . . . 6  |-  ( n  e.  D  ->  E. m  e.  om  n  =  suc  m )
41, 3bnj769 28792 . . . . 5  |-  ( ch 
->  E. m  e.  om  n  =  suc  m )
54bnj1196 28827 . . . 4  |-  ( ch 
->  E. m ( m  e.  om  /\  n  =  suc  m ) )
6 vex 2791 . . . . . 6  |-  n  e. 
_V
76sucex 4602 . . . . 5  |-  suc  n  e.  _V
87isseti 2794 . . . 4  |-  E. p  p  =  suc  n
95, 8jctir 524 . . 3  |-  ( ch 
->  ( E. m ( m  e.  om  /\  n  =  suc  m )  /\  E. p  p  =  suc  n ) )
10 exdistr 1847 . . . 4  |-  ( E. m E. p ( ( m  e.  om  /\  n  =  suc  m
)  /\  p  =  suc  n )  <->  E. m
( ( m  e. 
om  /\  n  =  suc  m )  /\  E. p  p  =  suc  n ) )
11 19.41v 1842 . . . 4  |-  ( E. m ( ( m  e.  om  /\  n  =  suc  m )  /\  E. p  p  =  suc  n )  <->  ( E. m ( m  e. 
om  /\  n  =  suc  m )  /\  E. p  p  =  suc  n ) )
1210, 11bitr2i 241 . . 3  |-  ( ( E. m ( m  e.  om  /\  n  =  suc  m )  /\  E. p  p  =  suc  n )  <->  E. m E. p ( ( m  e.  om  /\  n  =  suc  m )  /\  p  =  suc  n ) )
139, 12sylib 188 . 2  |-  ( ch 
->  E. m E. p
( ( m  e. 
om  /\  n  =  suc  m )  /\  p  =  suc  n ) )
14 bnj986.15 . . . 4  |-  ( ta  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )
15 df-3an 936 . . . 4  |-  ( ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n )  <-> 
( ( m  e. 
om  /\  n  =  suc  m )  /\  p  =  suc  n ) )
1614, 15bitri 240 . . 3  |-  ( ta  <->  ( ( m  e.  om  /\  n  =  suc  m
)  /\  p  =  suc  n ) )
17162exbii 1570 . 2  |-  ( E. m E. p ta  <->  E. m E. p ( ( m  e.  om  /\  n  =  suc  m
)  /\  p  =  suc  n ) )
1813, 17sylibr 203 1  |-  ( ch 
->  E. m E. p ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   E.wrex 2544    \ cdif 3149   (/)c0 3455   {csn 3640   suc csuc 4394   omcom 4656    Fn wfn 5250    /\ w-bnj17 28711
This theorem is referenced by:  bnj996  28987
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-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  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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