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Theorem bnj130 29222
Description: Technical lemma for bnj151 29225. 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
bnj130.1  |-  ( th  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
) )
bnj130.2  |-  ( ph'  <->  [. 1o  /  n ]. ph )
bnj130.3  |-  ( ps'  <->  [. 1o  /  n ]. ps )
bnj130.4  |-  ( th'  <->  [. 1o  /  n ]. th )
Assertion
Ref Expression
bnj130  |-  ( th'  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  1o  /\  ph'  /\  ps' ) ) )
Distinct variable groups:    A, n    R, n    f, n    x, n
Allowed substitution hints:    ph( x, f, n)    ps( x, f, n)    th( x, f, n)    A( x, f)    R( x, f)    ph'( x, f, n)    ps'( x, f, n)    th'( x, f, n)

Proof of Theorem bnj130
StepHypRef Expression
1 bnj130.1 . . 3  |-  ( th  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
) )
2 bnj105 29066 . . 3  |-  1o  e.  _V
31, 2bnj524 29082 . 2  |-  ( [. 1o  /  n ]. th  <->  [. 1o  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
) )
4 bnj130.4 . 2  |-  ( th'  <->  [. 1o  /  n ]. th )
52bnj90 29064 . . . . . . . . 9  |-  ( [. 1o  /  n ]. f  Fn  n  <->  f  Fn  1o )
65bicomi 193 . . . . . . . 8  |-  ( f  Fn  1o  <->  [. 1o  /  n ]. f  Fn  n
)
7 bnj130.2 . . . . . . . 8  |-  ( ph'  <->  [. 1o  /  n ]. ph )
8 bnj130.3 . . . . . . . 8  |-  ( ps'  <->  [. 1o  /  n ]. ps )
96, 7, 83anbi123i 1140 . . . . . . 7  |-  ( ( f  Fn  1o  /\  ph' 
/\  ps' )  <->  ( [. 1o  /  n ]. f  Fn  n  /\  [. 1o  /  n ]. ph  /\  [. 1o  /  n ]. ps ) )
10 sbc3ang 3062 . . . . . . . 8  |-  ( 1o  e.  _V  ->  ( [. 1o  /  n ]. ( f  Fn  n  /\  ph  /\  ps )  <->  (
[. 1o  /  n ]. f  Fn  n  /\  [. 1o  /  n ]. ph  /\  [. 1o  /  n ]. ps )
) )
112, 10ax-mp 8 . . . . . . 7  |-  ( [. 1o  /  n ]. (
f  Fn  n  /\  ph 
/\  ps )  <->  ( [. 1o  /  n ]. f  Fn  n  /\  [. 1o  /  n ]. ph  /\  [. 1o  /  n ]. ps ) )
129, 11bitr4i 243 . . . . . 6  |-  ( ( f  Fn  1o  /\  ph' 
/\  ps' )  <->  [. 1o  /  n ]. ( f  Fn  n  /\  ph  /\  ps ) )
1312eubii 2165 . . . . 5  |-  ( E! f ( f  Fn  1o  /\  ph'  /\  ps' )  <->  E! f [. 1o  /  n ]. ( f  Fn  n  /\  ph  /\  ps )
)
142bnj89 29063 . . . . 5  |-  ( [. 1o  /  n ]. E! f ( f  Fn  n  /\  ph  /\  ps )  <->  E! f [. 1o  /  n ]. ( f  Fn  n  /\  ph  /\ 
ps ) )
1513, 14bitr4i 243 . . . 4  |-  ( E! f ( f  Fn  1o  /\  ph'  /\  ps' )  <->  [. 1o  /  n ]. E! f ( f  Fn  n  /\  ph 
/\  ps ) )
1615imbi2i 303 . . 3  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  1o  /\  ph'  /\  ps' ) )  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  [. 1o  /  n ]. E! f ( f  Fn  n  /\  ph  /\  ps )
) )
17 nfv 1609 . . . . 5  |-  F/ n
( R  FrSe  A  /\  x  e.  A
)
1817sbc19.21g 3068 . . . 4  |-  ( 1o  e.  _V  ->  ( [. 1o  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
)  <->  ( ( R 
FrSe  A  /\  x  e.  A )  ->  [. 1o  /  n ]. E! f ( f  Fn  n  /\  ph  /\  ps )
) ) )
192, 18ax-mp 8 . . 3  |-  ( [. 1o  /  n ]. (
( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
)  <->  ( ( R 
FrSe  A  /\  x  e.  A )  ->  [. 1o  /  n ]. E! f ( f  Fn  n  /\  ph  /\  ps )
) )
2016, 19bitr4i 243 . 2  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  1o  /\  ph'  /\  ps' ) )  <->  [. 1o  /  n ]. ( ( R 
FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ph  /\  ps ) ) )
213, 4, 203bitr4i 268 1  |-  ( th'  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  1o  /\  ph'  /\  ps' ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1696   E!weu 2156   _Vcvv 2801   [.wsbc 3004    Fn wfn 5266   1oc1o 6488    FrSe w-bnj15 29033
This theorem is referenced by:  bnj151  29225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-pw 3640  df-sn 3659  df-suc 4414  df-fn 5274  df-1o 6495
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