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Theorem bnj207 29229
Description: Technical lemma for bnj852 29269. 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
bnj207.1  |-  ( ch  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
) )
bnj207.2  |-  ( ph'  <->  [. M  /  n ]. ph )
bnj207.3  |-  ( ps'  <->  [. M  /  n ]. ps )
bnj207.4  |-  ( ch'  <->  [. M  /  n ]. ch )
bnj207.5  |-  M  e. 
_V
Assertion
Ref Expression
bnj207  |-  ( ch'  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  M  /\  ph'  /\  ps' ) ) )
Distinct variable groups:    A, n    f, M    R, n    f, n   
x, n
Allowed substitution hints:    ph( x, f, n)    ps( x, f, n)    ch( x, f, n)    A( x, f)    R( x, f)    M( x, n)    ph'( x, f, n)    ps'( x, f, n)    ch'( x, f, n)

Proof of Theorem bnj207
StepHypRef Expression
1 bnj207.4 . 2  |-  ( ch'  <->  [. M  /  n ]. ch )
2 bnj207.1 . . . 4  |-  ( ch  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
) )
3 bnj207.5 . . . 4  |-  M  e. 
_V
42, 3bnj524 29082 . . 3  |-  ( [. M  /  n ]. ch  <->  [. M  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
) )
5 nfv 1609 . . . . . 6  |-  F/ n
( R  FrSe  A  /\  x  e.  A
)
65sbc19.21g 3068 . . . . 5  |-  ( M  e.  _V  ->  ( [. M  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
)  <->  ( ( R 
FrSe  A  /\  x  e.  A )  ->  [. M  /  n ]. E! f ( f  Fn  n  /\  ph  /\  ps )
) ) )
73, 6ax-mp 8 . . . 4  |-  ( [. M  /  n ]. (
( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
)  <->  ( ( R 
FrSe  A  /\  x  e.  A )  ->  [. M  /  n ]. E! f ( f  Fn  n  /\  ph  /\  ps )
) )
83bnj89 29063 . . . . . 6  |-  ( [. M  /  n ]. E! f ( f  Fn  n  /\  ph  /\  ps )  <->  E! f [. M  /  n ]. ( f  Fn  n  /\  ph  /\ 
ps ) )
93bnj90 29064 . . . . . . . . 9  |-  ( [. M  /  n ]. f  Fn  n  <->  f  Fn  M
)
109bicomi 193 . . . . . . . 8  |-  ( f  Fn  M  <->  [. M  /  n ]. f  Fn  n
)
11 bnj207.2 . . . . . . . 8  |-  ( ph'  <->  [. M  /  n ]. ph )
12 bnj207.3 . . . . . . . 8  |-  ( ps'  <->  [. M  /  n ]. ps )
1310, 11, 12, 3bnj206 29075 . . . . . . 7  |-  ( [. M  /  n ]. (
f  Fn  n  /\  ph 
/\  ps )  <->  ( f  Fn  M  /\  ph'  /\  ps' ) )
1413eubii 2165 . . . . . 6  |-  ( E! f [. M  /  n ]. ( f  Fn  n  /\  ph  /\  ps )  <->  E! f ( f  Fn  M  /\  ph'  /\  ps' ) )
158, 14bitri 240 . . . . 5  |-  ( [. M  /  n ]. E! f ( f  Fn  n  /\  ph  /\  ps )  <->  E! f ( f  Fn  M  /\  ph'  /\  ps' ) )
1615imbi2i 303 . . . 4  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  ->  [. M  /  n ]. E! f ( f  Fn  n  /\  ph 
/\  ps ) )  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  M  /\  ph'  /\  ps' ) ) )
177, 16bitri 240 . . 3  |-  ( [. M  /  n ]. (
( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
)  <->  ( ( R 
FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  M  /\  ph'  /\  ps' ) ) )
184, 17bitri 240 . 2  |-  ( [. M  /  n ]. ch  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  M  /\  ph'  /\  ps' ) ) )
191, 18bitri 240 1  |-  ( ch'  <->  (
( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  M  /\  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    FrSe w-bnj15 29033
This theorem is referenced by:  bnj600  29267  bnj908  29279
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
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-v 2803  df-sbc 3005  df-fn 5274
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