Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj523 Unicode version

Theorem bnj523 28919
Description: Technical lemma for bnj852 28953. 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
bnj523.1  |-  ( ph  <->  ( F `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj523.2  |-  ( ph'  <->  [. M  /  n ]. ph )
bnj523.3  |-  M  e. 
_V
Assertion
Ref Expression
bnj523  |-  ( ph'  <->  ( F `  (/) )  = 
pred ( X ,  A ,  R )
)
Distinct variable groups:    A, n    n, F    R, n    n, X
Allowed substitution hints:    ph( n)    M( n)    ph'( n)

Proof of Theorem bnj523
StepHypRef Expression
1 bnj523.2 . 2  |-  ( ph'  <->  [. M  /  n ]. ph )
2 bnj523.1 . . 3  |-  ( ph  <->  ( F `  (/) )  = 
pred ( X ,  A ,  R )
)
3 bnj523.3 . . 3  |-  M  e. 
_V
42, 3bnj524 28766 . 2  |-  ( [. M  /  n ]. ph  <->  [. M  /  n ]. ( F `  (/) )  =  pred ( X ,  A ,  R ) )
53bnj525 28767 . 2  |-  ( [. M  /  n ]. ( F `  (/) )  = 
pred ( X ,  A ,  R )  <->  ( F `  (/) )  = 
pred ( X ,  A ,  R )
)
61, 4, 53bitri 262 1  |-  ( ph'  <->  ( F `  (/) )  = 
pred ( X ,  A ,  R )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788   [.wsbc 2991   (/)c0 3455   ` cfv 5255    predc-bnj14 28713
This theorem is referenced by:  bnj600  28951  bnj908  28963  bnj934  28967
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992
  Copyright terms: Public domain W3C validator