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Theorem csbiedf 3010
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
csbiedf.1  |-  F/ x ph
csbiedf.2  |-  ( ph  -> 
F/_ x C )
csbiedf.3  |-  ( ph  ->  A  e.  V )
csbiedf.4  |-  ( (
ph  /\  x  =  A )  ->  B  =  C )
Assertion
Ref Expression
csbiedf  |-  ( ph  ->  [_ A  /  x ]_ B  =  C
)
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    B( x)    C( x)    V( x)

Proof of Theorem csbiedf
StepHypRef Expression
1 csbiedf.1 . . 3  |-  F/ x ph
2 csbiedf.4 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  B  =  C )
32ex 114 . . 3  |-  ( ph  ->  ( x  =  A  ->  B  =  C ) )
41, 3alrimi 1487 . 2  |-  ( ph  ->  A. x ( x  =  A  ->  B  =  C ) )
5 csbiedf.3 . . 3  |-  ( ph  ->  A  e.  V )
6 csbiedf.2 . . 3  |-  ( ph  -> 
F/_ x C )
7 csbiebt 3009 . . 3  |-  ( ( A  e.  V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
85, 6, 7syl2anc 408 . 2  |-  ( ph  ->  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
94, 8mpbid 146 1  |-  ( ph  ->  [_ A  /  x ]_ B  =  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1314    = wceq 1316   F/wnf 1421    e. wcel 1465   F/_wnfc 2245   [_csb 2975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883  df-csb 2976
This theorem is referenced by:  csbied  3016  csbie2t  3018
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