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Theorem csbiebt 2886
Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 2890.) (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 2566 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 spsbc 2775 . . . . 5  |-  ( A  e.  _V  ->  ( A. x ( x  =  A  ->  B  =  C )  ->  [. A  /  x ]. ( x  =  A  ->  B  =  C ) ) )
32adantr 261 . . . 4  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  ->  [. A  /  x ]. ( x  =  A  ->  B  =  C ) ) )
4 simpl 102 . . . . 5  |-  ( ( A  e.  _V  /\  F/_ x C )  ->  A  e.  _V )
5 biimt 230 . . . . . . 7  |-  ( x  =  A  ->  ( B  =  C  <->  ( x  =  A  ->  B  =  C ) ) )
6 csbeq1a 2860 . . . . . . . 8  |-  ( x  =  A  ->  B  =  [_ A  /  x ]_ B )
76eqeq1d 2048 . . . . . . 7  |-  ( x  =  A  ->  ( B  =  C  <->  [_ A  /  x ]_ B  =  C ) )
85, 7bitr3d 179 . . . . . 6  |-  ( x  =  A  ->  (
( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C
) )
98adantl 262 . . . . 5  |-  ( ( ( A  e.  _V  /\ 
F/_ x C )  /\  x  =  A )  ->  ( (
x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
10 nfv 1421 . . . . . 6  |-  F/ x  A  e.  _V
11 nfnfc1 2181 . . . . . 6  |-  F/ x F/_ x C
1210, 11nfan 1457 . . . . 5  |-  F/ x
( A  e.  _V  /\ 
F/_ x C )
13 nfcsb1v 2882 . . . . . . 7  |-  F/_ x [_ A  /  x ]_ B
1413a1i 9 . . . . . 6  |-  ( ( A  e.  _V  /\  F/_ x C )  ->  F/_ x [_ A  /  x ]_ B )
15 simpr 103 . . . . . 6  |-  ( ( A  e.  _V  /\  F/_ x C )  ->  F/_ x C )
1614, 15nfeqd 2192 . . . . 5  |-  ( ( A  e.  _V  /\  F/_ x C )  ->  F/ x [_ A  /  x ]_ B  =  C )
174, 9, 12, 16sbciedf 2798 . . . 4  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( [. A  /  x ]. ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C
) )
183, 17sylibd 138 . . 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 2192 . . . . . . 7  |-  ( F/_ x C  ->  F/ x [_ A  /  x ]_ B  =  C
)
2211, 21nfan1 1456 . . . . . 6  |-  F/ x
( F/_ x C  /\  [_ A  /  x ]_ B  =  C )
237biimprcd 149 . . . . . . 7  |-  ( [_ A  /  x ]_ B  =  C  ->  ( x  =  A  ->  B  =  C ) )
2423adantl 262 . . . . . 6  |-  ( (
F/_ x C  /\  [_ A  /  x ]_ B  =  C )  ->  ( x  =  A  ->  B  =  C ) )
2522, 24alrimi 1415 . . . . 5  |-  ( (
F/_ x C  /\  [_ A  /  x ]_ B  =  C )  ->  A. x ( x  =  A  ->  B  =  C ) )
2625ex 108 . . . 4  |-  ( F/_ x C  ->  ( [_ A  /  x ]_ B  =  C  ->  A. x
( x  =  A  ->  B  =  C ) ) )
2726adantl 262 . . 3  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( [_ A  /  x ]_ B  =  C  ->  A. x ( x  =  A  ->  B  =  C ) ) )
2818, 27impbid 120 . 2  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
291, 28sylan 267 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 97    <-> wb 98   A.wal 1241    = wceq 1243    e. wcel 1393   F/_wnfc 2165   _Vcvv 2557   [.wsbc 2764   [_csb 2852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853
This theorem is referenced by:  csbiedf  2887  csbieb  2888  csbiegf  2890
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