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Theorem bnj556 28977
Description: Technical lemma for bnj852 28998. 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 2919 . . . . 5  |-  p  e. 
_V
21bnj216 28805 . . . 4  |-  ( m  =  suc  p  ->  p  e.  m )
323anim3i 1141 . . 3  |-  ( ( m  e.  D  /\  n  =  suc  m  /\  m  =  suc  p )  ->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m ) )
43adantr 452 . 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 28778 . . 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 241 . 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 258 1  |-  ( et 
->  si )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   suc csuc 4543   omcom 4804    /\ w-bnj17 28756
This theorem is referenced by:  bnj557  28978  bnj561  28980  bnj562  28981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-un 3285  df-sn 3780  df-suc 4547  df-bnj17 28757
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