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Theorem bnj125 28574
Description: Technical lemma for bnj150 28578. 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 3152 . . 3  |-  ( [. F  /  f ]. ph'  <->  [. F  / 
f ]. [. 1o  /  n ]. ph )
4 bnj125.1 . . . . . 6  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
5 bnj105 28420 . . . . . 6  |-  1o  e.  _V
64, 5bnj91 28563 . . . . 5  |-  ( [. 1o  /  n ]. ph  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )
76sbcbii 3152 . . . 4  |-  ( [. F  /  f ]. [. 1o  /  n ]. ph  <->  [. F  / 
f ]. ( f `  (/) )  =  pred (
x ,  A ,  R ) )
8 bnj125.4 . . . . . 6  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
98bnj95 28566 . . . . 5  |-  F  e. 
_V
10 fveq1 5660 . . . . . 6  |-  ( f  =  F  ->  (
f `  (/) )  =  ( F `  (/) ) )
1110eqeq1d 2388 . . . . 5  |-  ( f  =  F  ->  (
( f `  (/) )  = 
pred ( x ,  A ,  R )  <-> 
( F `  (/) )  = 
pred ( x ,  A ,  R ) ) )
129, 11sbcie 3131 . . . 4  |-  ( [. F  /  f ]. (
f `  (/) )  = 
pred ( x ,  A ,  R )  <-> 
( F `  (/) )  = 
pred ( x ,  A ,  R ) )
137, 12bitri 241 . . 3  |-  ( [. F  /  f ]. [. 1o  /  n ]. ph  <->  ( F `  (/) )  =  pred ( x ,  A ,  R ) )
143, 13bitri 241 . 2  |-  ( [. F  /  f ]. ph'  <->  ( F `  (/) )  =  pred ( x ,  A ,  R ) )
151, 14bitri 241 1  |-  ( ph"  <->  ( F `  (/) )  =  pred ( x ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649   [.wsbc 3097   (/)c0 3564   {csn 3750   <.cop 3753   ` cfv 5387   1oc1o 6646    predc-bnj14 28383
This theorem is referenced by:  bnj150  28578  bnj153  28582
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
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 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-rex 2648  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-pw 3737  df-sn 3756  df-pr 3757  df-uni 3951  df-br 4147  df-suc 4521  df-iota 5351  df-fv 5395  df-1o 6653
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