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Theorem bnj986 29594
Description: Technical lemma for bnj69 29648. 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 29366 . . . . . 6  |-  ( n  e.  D  ->  E. m  e.  om  n  =  suc  m )
41, 3bnj769 29402 . . . . 5  |-  ( ch 
->  E. m  e.  om  n  =  suc  m )
54bnj1196 29435 . . . 4  |-  ( ch 
->  E. m ( m  e.  om  /\  n  =  suc  m ) )
6 vex 3081 . . . . . 6  |-  n  e. 
_V
76sucex 6643 . . . . 5  |-  suc  n  e.  _V
87isseti 3084 . . . 4  |-  E. p  p  =  suc  n
95, 8jctir 540 . . 3  |-  ( ch 
->  ( E. m ( m  e.  om  /\  n  =  suc  m )  /\  E. p  p  =  suc  n ) )
10 exdistr 1824 . . . 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 1819 . . . 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 253 . . 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 199 . 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 984 . . . 4  |-  ( ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n )  <-> 
( ( m  e. 
om  /\  n  =  suc  m )  /\  p  =  suc  n ) )
1614, 15bitri 252 . . 3  |-  ( ta  <->  ( ( m  e.  om  /\  n  =  suc  m
)  /\  p  =  suc  n ) )
17162exbii 1713 . 2  |-  ( E. m E. p ta  <->  E. m E. p ( ( m  e.  om  /\  n  =  suc  m
)  /\  p  =  suc  n ) )
1813, 17sylibr 215 1  |-  ( ch 
->  E. m E. p ta )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1659    e. wcel 1867   E.wrex 2774    \ cdif 3430   (/)c0 3758   {csn 3993   suc csuc 5435    Fn wfn 5587   omcom 6697    /\ w-bnj17 29320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-tr 4512  df-eprel 4756  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-om 6698  df-bnj17 29321
This theorem is referenced by:  bnj996  29595
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