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Theorem csbiebt 3450
Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3454.) (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 3118 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 spsbc 3340 . . . . 5  |-  ( A  e.  _V  ->  ( A. x ( x  =  A  ->  B  =  C )  ->  [. A  /  x ]. ( x  =  A  ->  B  =  C ) ) )
32adantr 465 . . . 4  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  ->  [. A  /  x ]. ( x  =  A  ->  B  =  C ) ) )
4 simpl 457 . . . . 5  |-  ( ( A  e.  _V  /\  F/_ x C )  ->  A  e.  _V )
5 biimt 335 . . . . . . 7  |-  ( x  =  A  ->  ( B  =  C  <->  ( x  =  A  ->  B  =  C ) ) )
6 csbeq1a 3439 . . . . . . . 8  |-  ( x  =  A  ->  B  =  [_ A  /  x ]_ B )
76eqeq1d 2459 . . . . . . 7  |-  ( x  =  A  ->  ( B  =  C  <->  [_ A  /  x ]_ B  =  C ) )
85, 7bitr3d 255 . . . . . 6  |-  ( x  =  A  ->  (
( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C
) )
98adantl 466 . . . . 5  |-  ( ( ( A  e.  _V  /\ 
F/_ x C )  /\  x  =  A )  ->  ( (
x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
10 nfv 1708 . . . . . 6  |-  F/ x  A  e.  _V
11 nfnfc1 2622 . . . . . 6  |-  F/ x F/_ x C
1210, 11nfan 1929 . . . . 5  |-  F/ x
( A  e.  _V  /\ 
F/_ x C )
13 nfcsb1v 3446 . . . . . . 7  |-  F/_ x [_ A  /  x ]_ B
1413a1i 11 . . . . . 6  |-  ( ( A  e.  _V  /\  F/_ x C )  ->  F/_ x [_ A  /  x ]_ B )
15 simpr 461 . . . . . 6  |-  ( ( A  e.  _V  /\  F/_ x C )  ->  F/_ x C )
1614, 15nfeqd 2626 . . . . 5  |-  ( ( A  e.  _V  /\  F/_ x C )  ->  F/ x [_ A  /  x ]_ B  =  C )
174, 9, 12, 16sbciedf 3363 . . . 4  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( [. A  /  x ]. ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C
) )
183, 17sylibd 214 . . 3  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  ->  [_ A  /  x ]_ B  =  C
) )
1913a1i 11 . . . . . . . 8  |-  ( F/_ x C  ->  F/_ x [_ A  /  x ]_ B )
20 id 22 . . . . . . . 8  |-  ( F/_ x C  ->  F/_ x C )
2119, 20nfeqd 2626 . . . . . . 7  |-  ( F/_ x C  ->  F/ x [_ A  /  x ]_ B  =  C
)
2211, 21nfan1 1928 . . . . . 6  |-  F/ x
( F/_ x C  /\  [_ A  /  x ]_ B  =  C )
237biimprcd 225 . . . . . . 7  |-  ( [_ A  /  x ]_ B  =  C  ->  ( x  =  A  ->  B  =  C ) )
2423adantl 466 . . . . . 6  |-  ( (
F/_ x C  /\  [_ A  /  x ]_ B  =  C )  ->  ( x  =  A  ->  B  =  C ) )
2522, 24alrimi 1878 . . . . 5  |-  ( (
F/_ x C  /\  [_ A  /  x ]_ B  =  C )  ->  A. x ( x  =  A  ->  B  =  C ) )
2625ex 434 . . . 4  |-  ( F/_ x C  ->  ( [_ A  /  x ]_ B  =  C  ->  A. x
( x  =  A  ->  B  =  C ) ) )
2726adantl 466 . . 3  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( [_ A  /  x ]_ B  =  C  ->  A. x ( x  =  A  ->  B  =  C ) ) )
2818, 27impbid 191 . 2  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
291, 28sylan 471 1  |-  ( ( A  e.  V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1393    = wceq 1395    e. wcel 1819   F/_wnfc 2605   _Vcvv 3109   [.wsbc 3327   [_csb 3430
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-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sbc 3328  df-csb 3431
This theorem is referenced by:  csbiedf  3451  csbieb  3452  csbiegf  3454
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