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Theorem csbieb 2888
Description: Bidirectional conversion between an implicit class substitution hypothesis  x  =  A  ->  B  =  C and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)
Hypotheses
Ref Expression
csbieb.1  |-  A  e. 
_V
csbieb.2  |-  F/_ x C
Assertion
Ref Expression
csbieb  |-  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    C( x)

Proof of Theorem csbieb
StepHypRef Expression
1 csbieb.1 . 2  |-  A  e. 
_V
2 csbieb.2 . 2  |-  F/_ x C
3 csbiebt 2886 . 2  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
41, 2, 3mp2an 402 1  |-  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98   A.wal 1241    = wceq 1243    e. wcel 1393   F/_wnfc 2165   _Vcvv 2557   [_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:  csbiebg  2889
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