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Theorem csbiebg 3406
Description: Bidirectional conversion between an implicit class substitution hypothesis  x  =  A  ->  B  =  C and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypothesis
Ref Expression
csbiebg.2  |-  F/_ x C
Assertion
Ref Expression
csbiebg  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C ) )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    C( x)    V( x)

Proof of Theorem csbiebg
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2465 . . . 4  |-  ( a  =  A  ->  (
x  =  a  <->  x  =  A ) )
21imbi1d 317 . . 3  |-  ( a  =  A  ->  (
( x  =  a  ->  B  =  C )  <->  ( x  =  A  ->  B  =  C ) ) )
32albidv 1680 . 2  |-  ( a  =  A  ->  ( A. x ( x  =  a  ->  B  =  C )  <->  A. x
( x  =  A  ->  B  =  C ) ) )
4 csbeq1 3386 . . 3  |-  ( a  =  A  ->  [_ a  /  x ]_ B  = 
[_ A  /  x ]_ B )
54eqeq1d 2453 . 2  |-  ( a  =  A  ->  ( [_ a  /  x ]_ B  =  C  <->  [_ A  /  x ]_ B  =  C )
)
6 vex 3068 . . 3  |-  a  e. 
_V
7 csbiebg.2 . . 3  |-  F/_ x C
86, 7csbieb 3405 . 2  |-  ( A. x ( x  =  a  ->  B  =  C )  <->  [_ a  /  x ]_ B  =  C )
93, 5, 8vtoclbg 3124 1  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1368    = wceq 1370    e. wcel 1758   F/_wnfc 2597   [_csb 3383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-v 3067  df-sbc 3282  df-csb 3384
This theorem is referenced by:  cdlemefrs29bpre0  34343
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