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Theorem bnj556 29720
Description: Technical lemma for bnj852 29741. 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
bnj556.18  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
bnj556.19  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
Assertion
Ref Expression
bnj556  |-  ( et 
->  si )

Proof of Theorem bnj556
StepHypRef Expression
1 vex 3083 . . . . 5  |-  p  e. 
_V
21bnj216 29549 . . . 4  |-  ( m  =  suc  p  ->  p  e.  m )
323anim3i 1193 . . 3  |-  ( ( m  e.  D  /\  n  =  suc  m  /\  m  =  suc  p )  ->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m ) )
43adantr 466 . 2  |-  ( ( ( m  e.  D  /\  n  =  suc  m  /\  m  =  suc  p )  /\  p  e.  om )  ->  (
m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
5 bnj556.19 . . 3  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
6 bnj258 29522 . . 3  |-  ( ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p )  <->  ( (
m  e.  D  /\  n  =  suc  m  /\  m  =  suc  p )  /\  p  e.  om ) )
75, 6bitri 252 . 2  |-  ( et  <->  ( ( m  e.  D  /\  n  =  suc  m  /\  m  =  suc  p )  /\  p  e.  om ) )
8 bnj556.18 . 2  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
94, 7, 83imtr4i 269 1  |-  ( et 
->  si )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   suc csuc 5444   omcom 6707    /\ w-bnj17 29500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-un 3441  df-sn 3999  df-suc 5448  df-bnj17 29501
This theorem is referenced by:  bnj557  29721  bnj561  29723  bnj562  29724
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