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Theorem csbiebt 3117
Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3121.) (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 2796 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 spsbc 3003 . . . . 5  |-  ( A  e.  _V  ->  ( A. x ( x  =  A  ->  B  =  C )  ->  [. A  /  x ]. ( x  =  A  ->  B  =  C ) ) )
32adantr 451 . . . 4  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  ->  [. A  /  x ]. ( x  =  A  ->  B  =  C ) ) )
4 simpl 443 . . . . 5  |-  ( ( A  e.  _V  /\  F/_ x C )  ->  A  e.  _V )
5 biimt 325 . . . . . . 7  |-  ( x  =  A  ->  ( B  =  C  <->  ( x  =  A  ->  B  =  C ) ) )
6 csbeq1a 3089 . . . . . . . 8  |-  ( x  =  A  ->  B  =  [_ A  /  x ]_ B )
76eqeq1d 2291 . . . . . . 7  |-  ( x  =  A  ->  ( B  =  C  <->  [_ A  /  x ]_ B  =  C ) )
85, 7bitr3d 246 . . . . . 6  |-  ( x  =  A  ->  (
( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C
) )
98adantl 452 . . . . 5  |-  ( ( ( A  e.  _V  /\ 
F/_ x C )  /\  x  =  A )  ->  ( (
x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
10 nfv 1605 . . . . . 6  |-  F/ x  A  e.  _V
11 nfnfc1 2422 . . . . . 6  |-  F/ x F/_ x C
1210, 11nfan 1771 . . . . 5  |-  F/ x
( A  e.  _V  /\ 
F/_ x C )
13 nfcsb1v 3113 . . . . . . 7  |-  F/_ x [_ A  /  x ]_ B
1413a1i 10 . . . . . 6  |-  ( ( A  e.  _V  /\  F/_ x C )  ->  F/_ x [_ A  /  x ]_ B )
15 simpr 447 . . . . . 6  |-  ( ( A  e.  _V  /\  F/_ x C )  ->  F/_ x C )
1614, 15nfeqd 2433 . . . . 5  |-  ( ( A  e.  _V  /\  F/_ x C )  ->  F/ x [_ A  /  x ]_ B  =  C )
174, 9, 12, 16sbciedf 3026 . . . 4  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( [. A  /  x ]. ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C
) )
183, 17sylibd 205 . . 3  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  ->  [_ A  /  x ]_ B  =  C
) )
1913a1i 10 . . . . . . . 8  |-  ( F/_ x C  ->  F/_ x [_ A  /  x ]_ B )
20 id 19 . . . . . . . 8  |-  ( F/_ x C  ->  F/_ x C )
2119, 20nfeqd 2433 . . . . . . 7  |-  ( F/_ x C  ->  F/ x [_ A  /  x ]_ B  =  C
)
2211, 21nfan1 1822 . . . . . 6  |-  F/ x
( F/_ x C  /\  [_ A  /  x ]_ B  =  C )
237biimprcd 216 . . . . . . 7  |-  ( [_ A  /  x ]_ B  =  C  ->  ( x  =  A  ->  B  =  C ) )
2423adantl 452 . . . . . 6  |-  ( (
F/_ x C  /\  [_ A  /  x ]_ B  =  C )  ->  ( x  =  A  ->  B  =  C ) )
2522, 24alrimi 1745 . . . . 5  |-  ( (
F/_ x C  /\  [_ A  /  x ]_ B  =  C )  ->  A. x ( x  =  A  ->  B  =  C ) )
2625ex 423 . . . 4  |-  ( F/_ x C  ->  ( [_ A  /  x ]_ B  =  C  ->  A. x
( x  =  A  ->  B  =  C ) ) )
2726adantl 452 . . 3  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( [_ A  /  x ]_ B  =  C  ->  A. x ( x  =  A  ->  B  =  C ) ) )
2818, 27impbid 183 . 2  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
291, 28sylan 457 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    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   F/_wnfc 2406   _Vcvv 2788   [.wsbc 2991   [_csb 3081
This theorem is referenced by:  csbiedf  3118  csbieb  3119  csbiegf  3121
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992  df-csb 3082
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