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Theorem bnj125 33410
Description: Technical lemma for bnj150 33414. 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
bnj125.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj125.2  |-  ( ph'  <->  [. 1o  /  n ]. ph )
bnj125.3  |-  ( ph"  <->  [. F  / 
f ]. ph' )
bnj125.4  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
Assertion
Ref Expression
bnj125  |-  ( ph"  <->  ( F `  (/) )  =  pred ( x ,  A ,  R ) )
Distinct variable groups:    A, f, n    f, F    R, f, n    x, f, n
Allowed substitution hints:    ph( x, f, n)    A( x)    R( x)    F( x, n)    ph'( x, f, n)   
ph"( x, f, n)

Proof of Theorem bnj125
StepHypRef Expression
1 bnj125.3 . 2  |-  ( ph"  <->  [. F  / 
f ]. ph' )
2 bnj125.2 . . . 4  |-  ( ph'  <->  [. 1o  /  n ]. ph )
32sbcbii 3396 . . 3  |-  ( [. F  /  f ]. ph'  <->  [. F  / 
f ]. [. 1o  /  n ]. ph )
4 bnj125.1 . . . . . 6  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
5 bnj105 33258 . . . . . 6  |-  1o  e.  _V
64, 5bnj91 33399 . . . . 5  |-  ( [. 1o  /  n ]. ph  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )
76sbcbii 3396 . . . 4  |-  ( [. F  /  f ]. [. 1o  /  n ]. ph  <->  [. F  / 
f ]. ( f `  (/) )  =  pred (
x ,  A ,  R ) )
8 bnj125.4 . . . . . 6  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
98bnj95 33402 . . . . 5  |-  F  e. 
_V
10 fveq1 5871 . . . . . 6  |-  ( f  =  F  ->  (
f `  (/) )  =  ( F `  (/) ) )
1110eqeq1d 2469 . . . . 5  |-  ( f  =  F  ->  (
( f `  (/) )  = 
pred ( x ,  A ,  R )  <-> 
( F `  (/) )  = 
pred ( x ,  A ,  R ) ) )
129, 11sbcie 3371 . . . 4  |-  ( [. F  /  f ]. (
f `  (/) )  = 
pred ( x ,  A ,  R )  <-> 
( F `  (/) )  = 
pred ( x ,  A ,  R ) )
137, 12bitri 249 . . 3  |-  ( [. F  /  f ]. [. 1o  /  n ]. ph  <->  ( F `  (/) )  =  pred ( x ,  A ,  R ) )
143, 13bitri 249 . 2  |-  ( [. F  /  f ]. ph'  <->  ( F `  (/) )  =  pred ( x ,  A ,  R ) )
151, 14bitri 249 1  |-  ( ph"  <->  ( F `  (/) )  =  pred ( x ,  A ,  R ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379   [.wsbc 3336   (/)c0 3790   {csn 4033   <.cop 4039   ` cfv 5594   1oc1o 7135    predc-bnj14 33221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rex 2823  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-pw 4018  df-sn 4034  df-pr 4036  df-uni 4252  df-br 4454  df-suc 4890  df-iota 5557  df-fv 5602  df-1o 7142
This theorem is referenced by:  bnj150  33414  bnj153  33418
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