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Theorem bnj1021 29823
Description: Technical lemma for bnj69 29867. 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
bnj1021.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj1021.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj1021.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj1021.4  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
)  /\  z  e.  pred ( y ,  A ,  R ) ) )
bnj1021.5  |-  ( ta  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )
bnj1021.6  |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i ) ) )
bnj1021.13  |-  D  =  ( om  \  { (/)
} )
bnj1021.14  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
Assertion
Ref Expression
bnj1021  |-  E. f E. n E. i E. m ( th  ->  ( th  /\  ch  /\  et  /\  E. p ta ) )
Distinct variable groups:    A, f,
i, n, y    D, i    R, f, i, n, y    f, X, i, n, y    ch, m, p    et, m, p    th, f,
i, n    ph, i    m, n, th, p
Allowed substitution hints:    ph( y, z, f, m, n, p)    ps( y, z, f, i, m, n, p)    ch( y, z, f, i, n)    th( y, z)    ta( y,
z, f, i, m, n, p)    et( y,
z, f, i, n)    A( z, m, p)    B( y, z, f, i, m, n, p)    D( y,
z, f, m, n, p)    R( z, m, p)    X( z, m, p)

Proof of Theorem bnj1021
StepHypRef Expression
1 bnj1021.1 . . . 4  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
2 bnj1021.2 . . . 4  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 bnj1021.3 . . . 4  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
4 bnj1021.4 . . . 4  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R
)  /\  z  e.  pred ( y ,  A ,  R ) ) )
5 bnj1021.5 . . . 4  |-  ( ta  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )
6 bnj1021.6 . . . 4  |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i ) ) )
7 bnj1021.13 . . . 4  |-  D  =  ( om  \  { (/)
} )
8 bnj1021.14 . . . 4  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
91, 2, 3, 4, 5, 6, 7, 8bnj996 29814 . . 3  |-  E. f E. n E. i E. m E. p ( th  ->  ( ch  /\ 
ta  /\  et )
)
10 anclb 554 . . . . . 6  |-  ( ( th  ->  ( ch  /\ 
ta  /\  et )
)  <->  ( th  ->  ( th  /\  ( ch 
/\  ta  /\  et ) ) ) )
11 bnj252 29556 . . . . . . 7  |-  ( ( th  /\  ch  /\  ta  /\  et )  <->  ( th  /\  ( ch  /\  ta  /\  et ) ) )
1211imbi2i 318 . . . . . 6  |-  ( ( th  ->  ( th  /\  ch  /\  ta  /\  et ) )  <->  ( th  ->  ( th  /\  ( ch  /\  ta  /\  et ) ) ) )
1310, 12bitr4i 260 . . . . 5  |-  ( ( th  ->  ( ch  /\ 
ta  /\  et )
)  <->  ( th  ->  ( th  /\  ch  /\  ta  /\  et ) ) )
14132exbii 1729 . . . 4  |-  ( E. m E. p ( th  ->  ( ch  /\ 
ta  /\  et )
)  <->  E. m E. p
( th  ->  ( th  /\  ch  /\  ta  /\  et ) ) )
15143exbii 1730 . . 3  |-  ( E. f E. n E. i E. m E. p
( th  ->  ( ch  /\  ta  /\  et ) )  <->  E. f E. n E. i E. m E. p ( th  ->  ( th  /\  ch  /\  ta  /\  et ) ) )
169, 15mpbi 213 . 2  |-  E. f E. n E. i E. m E. p ( th  ->  ( th  /\  ch  /\  ta  /\  et ) )
17 19.37v 1836 . . . . 5  |-  ( E. p ( th  ->  ( th  /\  ch  /\  ta  /\  et ) )  <-> 
( th  ->  E. p
( th  /\  ch  /\ 
ta  /\  et )
) )
18 bnj1019 29639 . . . . . 6  |-  ( E. p ( th  /\  ch  /\  ta  /\  et ) 
<->  ( th  /\  ch  /\  et  /\  E. p ta ) )
1918imbi2i 318 . . . . 5  |-  ( ( th  ->  E. p
( th  /\  ch  /\ 
ta  /\  et )
)  <->  ( th  ->  ( th  /\  ch  /\  et  /\  E. p ta ) ) )
2017, 19bitri 257 . . . 4  |-  ( E. p ( th  ->  ( th  /\  ch  /\  ta  /\  et ) )  <-> 
( th  ->  ( th  /\  ch  /\  et  /\  E. p ta )
) )
21202exbii 1729 . . 3  |-  ( E. i E. m E. p ( th  ->  ( th  /\  ch  /\  ta  /\  et ) )  <->  E. i E. m ( th  ->  ( th  /\  ch  /\  et  /\  E. p ta ) ) )
22212exbii 1729 . 2  |-  ( E. f E. n E. i E. m E. p
( th  ->  ( th  /\  ch  /\  ta  /\  et ) )  <->  E. f E. n E. i E. m ( th  ->  ( th  /\  ch  /\  et  /\  E. p ta ) ) )
2316, 22mpbi 213 1  |-  E. f E. n E. i E. m ( th  ->  ( th  /\  ch  /\  et  /\  E. p ta ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1454   E.wex 1673    e. wcel 1897   {cab 2447   A.wral 2748   E.wrex 2749    \ cdif 3412   (/)c0 3742   {csn 3979   U_ciun 4291   suc csuc 5443    Fn wfn 5595   ` cfv 5600   omcom 6718    /\ w-bnj17 29539    predc-bnj14 29541    FrSe w-bnj15 29545    trClc-bnj18 29547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-tr 4511  df-eprel 4763  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-fn 5603  df-om 6719  df-bnj17 29540  df-bnj18 29548
This theorem is referenced by:  bnj907  29824
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