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Theorem csbiebt 2951
Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 2955.) (Contributed by NM, 11-Nov-2005.)
Assertion
Ref Expression
csbiebt  |-  ( ( A  e.  V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    C( x)    V( x)

Proof of Theorem csbiebt
StepHypRef Expression
1 elex 2619 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 spsbc 2835 . . . . 5  |-  ( A  e.  _V  ->  ( A. x ( x  =  A  ->  B  =  C )  ->  [. A  /  x ]. ( x  =  A  ->  B  =  C ) ) )
32adantr 270 . . . 4  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  ->  [. A  /  x ]. ( x  =  A  ->  B  =  C ) ) )
4 simpl 107 . . . . 5  |-  ( ( A  e.  _V  /\  F/_ x C )  ->  A  e.  _V )
5 biimt 239 . . . . . . 7  |-  ( x  =  A  ->  ( B  =  C  <->  ( x  =  A  ->  B  =  C ) ) )
6 csbeq1a 2925 . . . . . . . 8  |-  ( x  =  A  ->  B  =  [_ A  /  x ]_ B )
76eqeq1d 2091 . . . . . . 7  |-  ( x  =  A  ->  ( B  =  C  <->  [_ A  /  x ]_ B  =  C ) )
85, 7bitr3d 188 . . . . . 6  |-  ( x  =  A  ->  (
( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C
) )
98adantl 271 . . . . 5  |-  ( ( ( A  e.  _V  /\ 
F/_ x C )  /\  x  =  A )  ->  ( (
x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
10 nfv 1462 . . . . . 6  |-  F/ x  A  e.  _V
11 nfnfc1 2226 . . . . . 6  |-  F/ x F/_ x C
1210, 11nfan 1498 . . . . 5  |-  F/ x
( A  e.  _V  /\ 
F/_ x C )
13 nfcsb1v 2947 . . . . . . 7  |-  F/_ x [_ A  /  x ]_ B
1413a1i 9 . . . . . 6  |-  ( ( A  e.  _V  /\  F/_ x C )  ->  F/_ x [_ A  /  x ]_ B )
15 simpr 108 . . . . . 6  |-  ( ( A  e.  _V  /\  F/_ x C )  ->  F/_ x C )
1614, 15nfeqd 2237 . . . . 5  |-  ( ( A  e.  _V  /\  F/_ x C )  ->  F/ x [_ A  /  x ]_ B  =  C )
174, 9, 12, 16sbciedf 2858 . . . 4  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( [. A  /  x ]. ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C
) )
183, 17sylibd 147 . . 3  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  ->  [_ A  /  x ]_ B  =  C
) )
1913a1i 9 . . . . . . . 8  |-  ( F/_ x C  ->  F/_ x [_ A  /  x ]_ B )
20 id 19 . . . . . . . 8  |-  ( F/_ x C  ->  F/_ x C )
2119, 20nfeqd 2237 . . . . . . 7  |-  ( F/_ x C  ->  F/ x [_ A  /  x ]_ B  =  C
)
2211, 21nfan1 1497 . . . . . 6  |-  F/ x
( F/_ x C  /\  [_ A  /  x ]_ B  =  C )
237biimprcd 158 . . . . . . 7  |-  ( [_ A  /  x ]_ B  =  C  ->  ( x  =  A  ->  B  =  C ) )
2423adantl 271 . . . . . 6  |-  ( (
F/_ x C  /\  [_ A  /  x ]_ B  =  C )  ->  ( x  =  A  ->  B  =  C ) )
2522, 24alrimi 1456 . . . . 5  |-  ( (
F/_ x C  /\  [_ A  /  x ]_ B  =  C )  ->  A. x ( x  =  A  ->  B  =  C ) )
2625ex 113 . . . 4  |-  ( F/_ x C  ->  ( [_ A  /  x ]_ B  =  C  ->  A. x
( x  =  A  ->  B  =  C ) ) )
2726adantl 271 . . 3  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( [_ A  /  x ]_ B  =  C  ->  A. x ( x  =  A  ->  B  =  C ) ) )
2818, 27impbid 127 . 2  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
291, 28sylan 277 1  |-  ( ( A  e.  V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285    e. wcel 1434   F/_wnfc 2210   _Vcvv 2610   [.wsbc 2824   [_csb 2917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-sbc 2825  df-csb 2918
This theorem is referenced by:  csbiedf  2952  csbieb  2953  csbiegf  2955
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