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Theorem bnj130 34054
Description: Technical lemma for bnj151 34057. 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 )
) )
21sbcbii 3387 . 2  |-  ( [. 1o  /  n ]. th  <->  [. 1o  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
) )
3 bnj130.4 . 2  |-  ( th'  <->  [. 1o  /  n ]. th )
4 bnj105 33899 . . . . . . . . . 10  |-  1o  e.  _V
54bnj90 33897 . . . . . . . . 9  |-  ( [. 1o  /  n ]. f  Fn  n  <->  f  Fn  1o )
65bicomi 202 . . . . . . . 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 1185 . . . . . . 7  |-  ( ( f  Fn  1o  /\  ph' 
/\  ps' )  <->  ( [. 1o  /  n ]. f  Fn  n  /\  [. 1o  /  n ]. ph  /\  [. 1o  /  n ]. ps ) )
10 sbc3an 3390 . . . . . . 7  |-  ( [. 1o  /  n ]. (
f  Fn  n  /\  ph 
/\  ps )  <->  ( [. 1o  /  n ]. f  Fn  n  /\  [. 1o  /  n ]. ph  /\  [. 1o  /  n ]. ps ) )
119, 10bitr4i 252 . . . . . 6  |-  ( ( f  Fn  1o  /\  ph' 
/\  ps' )  <->  [. 1o  /  n ]. ( f  Fn  n  /\  ph  /\  ps ) )
1211eubii 2307 . . . . 5  |-  ( E! f ( f  Fn  1o  /\  ph'  /\  ps' )  <->  E! f [. 1o  /  n ]. ( f  Fn  n  /\  ph  /\  ps )
)
134bnj89 33896 . . . . 5  |-  ( [. 1o  /  n ]. E! f ( f  Fn  n  /\  ph  /\  ps )  <->  E! f [. 1o  /  n ]. ( f  Fn  n  /\  ph  /\ 
ps ) )
1412, 13bitr4i 252 . . . 4  |-  ( E! f ( f  Fn  1o  /\  ph'  /\  ps' )  <->  [. 1o  /  n ]. E! f ( f  Fn  n  /\  ph 
/\  ps ) )
1514imbi2i 312 . . 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 )
) )
16 nfv 1708 . . . . 5  |-  F/ n
( R  FrSe  A  /\  x  e.  A
)
1716sbc19.21g 3398 . . . 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 )
) ) )
184, 17ax-mp 5 . . 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 )
) )
1915, 18bitr4i 252 . 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 ) ) )
202, 3, 193bitr4i 277 1  |-  ( th'  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  1o  /\  ph'  /\  ps' ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1819   E!weu 2283   _Vcvv 3109   [.wsbc 3327    Fn wfn 5589   1oc1o 7141    FrSe w-bnj15 33866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-pw 4017  df-sn 4033  df-suc 4893  df-fn 5597  df-1o 7148
This theorem is referenced by:  bnj151  34057
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