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Theorem bnj986 29226
Description: Technical lemma for bnj69 29280. 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 28997 . . . . . 6  |-  ( n  e.  D  ->  E. m  e.  om  n  =  suc  m )
41, 3bnj769 29032 . . . . 5  |-  ( ch 
->  E. m  e.  om  n  =  suc  m )
54bnj1196 29067 . . . 4  |-  ( ch 
->  E. m ( m  e.  om  /\  n  =  suc  m ) )
6 vex 2951 . . . . . 6  |-  n  e. 
_V
76sucex 4783 . . . . 5  |-  suc  n  e.  _V
87isseti 2954 . . . 4  |-  E. p  p  =  suc  n
95, 8jctir 525 . . 3  |-  ( ch 
->  ( E. m ( m  e.  om  /\  n  =  suc  m )  /\  E. p  p  =  suc  n ) )
10 exdistr 1929 . . . 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 1924 . . . 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 242 . . 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 189 . 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 938 . . . 4  |-  ( ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n )  <-> 
( ( m  e. 
om  /\  n  =  suc  m )  /\  p  =  suc  n ) )
1614, 15bitri 241 . . 3  |-  ( ta  <->  ( ( m  e.  om  /\  n  =  suc  m
)  /\  p  =  suc  n ) )
17162exbii 1593 . 2  |-  ( E. m E. p ta  <->  E. m E. p ( ( m  e.  om  /\  n  =  suc  m
)  /\  p  =  suc  n ) )
1813, 17sylibr 204 1  |-  ( ch 
->  E. m E. p ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   E.wrex 2698    \ cdif 3309   (/)c0 3620   {csn 3806   suc csuc 4575   omcom 4837    Fn wfn 5441    /\ w-bnj17 28951
This theorem is referenced by:  bnj996  29227
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-bnj17 28952
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