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Theorem csbiedf 3231
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 424 . . 3  |-  ( ph  ->  ( x  =  A  ->  B  =  C ) )
41, 3alrimi 1773 . 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 3230 . . 3  |-  ( ( A  e.  V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
85, 6, 7syl2anc 643 . 2  |-  ( ph  ->  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
94, 8mpbid 202 1  |-  ( ph  ->  [_ A  /  x ]_ B  =  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   F/wnf 1550    = wceq 1649    e. wcel 1717   F/_wnfc 2510   [_csb 3194
This theorem is referenced by:  csbied  3236  csbie2t  3238  natpropd  14100  fucpropd  14101  sumsnd  27365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-sbc 3105  df-csb 3195
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