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Theorem bnj523 29746
Description: Technical lemma for bnj852 29780. 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 )
)
32sbcbii 3334 . 2  |-  ( [. M  /  n ]. ph  <->  [. M  /  n ]. ( F `  (/) )  =  pred ( X ,  A ,  R ) )
4 bnj523.3 . . 3  |-  M  e. 
_V
54bnj525 29595 . 2  |-  ( [. M  /  n ]. ( F `  (/) )  = 
pred ( X ,  A ,  R )  <->  ( F `  (/) )  = 
pred ( X ,  A ,  R )
)
61, 3, 53bitri 279 1  |-  ( ph'  <->  ( F `  (/) )  = 
pred ( X ,  A ,  R )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    = wceq 1454    e. wcel 1897   _Vcvv 3056   [.wsbc 3278   (/)c0 3742   ` cfv 5600    predc-bnj14 29541
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-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-v 3058  df-sbc 3279
This theorem is referenced by:  bnj600  29778  bnj908  29790  bnj934  29794
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