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Theorem csbiedf 2887
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 108 . . 3  |-  ( ph  ->  ( x  =  A  ->  B  =  C ) )
41, 3alrimi 1415 . 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 2886 . . 3  |-  ( ( A  e.  V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
85, 6, 7syl2anc 391 . 2  |-  ( ph  ->  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
94, 8mpbid 135 1  |-  ( ph  ->  [_ A  /  x ]_ B  =  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98   A.wal 1241    = wceq 1243   F/wnf 1349    e. wcel 1393   F/_wnfc 2165   [_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:  csbied  2892  csbie2t  2894
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