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Theorem csbiebt 3009
Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3013.) (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 2671 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 spsbc 2893 . . . . 5  |-  ( A  e.  _V  ->  ( A. x ( x  =  A  ->  B  =  C )  ->  [. A  /  x ]. ( x  =  A  ->  B  =  C ) ) )
32adantr 274 . . . 4  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  ->  [. A  /  x ]. ( x  =  A  ->  B  =  C ) ) )
4 simpl 108 . . . . 5  |-  ( ( A  e.  _V  /\  F/_ x C )  ->  A  e.  _V )
5 biimt 240 . . . . . . 7  |-  ( x  =  A  ->  ( B  =  C  <->  ( x  =  A  ->  B  =  C ) ) )
6 csbeq1a 2983 . . . . . . . 8  |-  ( x  =  A  ->  B  =  [_ A  /  x ]_ B )
76eqeq1d 2126 . . . . . . 7  |-  ( x  =  A  ->  ( B  =  C  <->  [_ A  /  x ]_ B  =  C ) )
85, 7bitr3d 189 . . . . . 6  |-  ( x  =  A  ->  (
( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C
) )
98adantl 275 . . . . 5  |-  ( ( ( A  e.  _V  /\ 
F/_ x C )  /\  x  =  A )  ->  ( (
x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
10 nfv 1493 . . . . . 6  |-  F/ x  A  e.  _V
11 nfnfc1 2261 . . . . . 6  |-  F/ x F/_ x C
1210, 11nfan 1529 . . . . 5  |-  F/ x
( A  e.  _V  /\ 
F/_ x C )
13 nfcsb1v 3005 . . . . . . 7  |-  F/_ x [_ A  /  x ]_ B
1413a1i 9 . . . . . 6  |-  ( ( A  e.  _V  /\  F/_ x C )  ->  F/_ x [_ A  /  x ]_ B )
15 simpr 109 . . . . . 6  |-  ( ( A  e.  _V  /\  F/_ x C )  ->  F/_ x C )
1614, 15nfeqd 2273 . . . . 5  |-  ( ( A  e.  _V  /\  F/_ x C )  ->  F/ x [_ A  /  x ]_ B  =  C )
174, 9, 12, 16sbciedf 2916 . . . 4  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( [. A  /  x ]. ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C
) )
183, 17sylibd 148 . . 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 2273 . . . . . . 7  |-  ( F/_ x C  ->  F/ x [_ A  /  x ]_ B  =  C
)
2211, 21nfan1 1528 . . . . . 6  |-  F/ x
( F/_ x C  /\  [_ A  /  x ]_ B  =  C )
237biimprcd 159 . . . . . . 7  |-  ( [_ A  /  x ]_ B  =  C  ->  ( x  =  A  ->  B  =  C ) )
2423adantl 275 . . . . . 6  |-  ( (
F/_ x C  /\  [_ A  /  x ]_ B  =  C )  ->  ( x  =  A  ->  B  =  C ) )
2522, 24alrimi 1487 . . . . 5  |-  ( (
F/_ x C  /\  [_ A  /  x ]_ B  =  C )  ->  A. x ( x  =  A  ->  B  =  C ) )
2625ex 114 . . . 4  |-  ( F/_ x C  ->  ( [_ A  /  x ]_ B  =  C  ->  A. x
( x  =  A  ->  B  =  C ) ) )
2726adantl 275 . . 3  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( [_ A  /  x ]_ B  =  C  ->  A. x ( x  =  A  ->  B  =  C ) ) )
2818, 27impbid 128 . 2  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
291, 28sylan 281 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 103    <-> wb 104   A.wal 1314    = wceq 1316    e. wcel 1465   F/_wnfc 2245   _Vcvv 2660   [.wsbc 2882   [_csb 2975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883  df-csb 2976
This theorem is referenced by:  csbiedf  3010  csbieb  3011  csbiegf  3013
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