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Theorem csbieb 3098
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 3096 . 2  |-  ( ( A  e.  _V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
41, 2, 3mp2an 426 1  |-  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1351    = wceq 1353    e. wcel 2148   F/_wnfc 2306   _Vcvv 2737   [_csb 3057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963  df-csb 3058
This theorem is referenced by:  csbiebg  3099
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