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Theorem bnj986 29815
Description: Technical lemma for bnj69 29869. 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 29587 . . . . . 6  |-  ( n  e.  D  ->  E. m  e.  om  n  =  suc  m )
41, 3bnj769 29623 . . . . 5  |-  ( ch 
->  E. m  e.  om  n  =  suc  m )
54bnj1196 29656 . . . 4  |-  ( ch 
->  E. m ( m  e.  om  /\  n  =  suc  m ) )
6 vex 3060 . . . . . 6  |-  n  e. 
_V
76sucex 6670 . . . . 5  |-  suc  n  e.  _V
87isseti 3063 . . . 4  |-  E. p  p  =  suc  n
95, 8jctir 545 . . 3  |-  ( ch 
->  ( E. m ( m  e.  om  /\  n  =  suc  m )  /\  E. p  p  =  suc  n ) )
10 exdistr 1846 . . . 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 1841 . . . 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 258 . . 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 201 . 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 993 . . . 4  |-  ( ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n )  <-> 
( ( m  e. 
om  /\  n  =  suc  m )  /\  p  =  suc  n ) )
1614, 15bitri 257 . . 3  |-  ( ta  <->  ( ( m  e.  om  /\  n  =  suc  m
)  /\  p  =  suc  n ) )
17162exbii 1730 . 2  |-  ( E. m E. p ta  <->  E. m E. p ( ( m  e.  om  /\  n  =  suc  m
)  /\  p  =  suc  n ) )
1813, 17sylibr 217 1  |-  ( ch 
->  E. m E. p ta )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455   E.wex 1674    e. wcel 1898   E.wrex 2750    \ cdif 3413   (/)c0 3743   {csn 3980   suc csuc 5448    Fn wfn 5600   omcom 6724    /\ w-bnj17 29541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-tr 4514  df-eprel 4767  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-om 6725  df-bnj17 29542
This theorem is referenced by:  bnj996  29816
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