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Theorem csbiebg 3290
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 2445 . . . 4  |-  ( a  =  A  ->  (
x  =  a  <->  x  =  A ) )
21imbi1d 309 . . 3  |-  ( a  =  A  ->  (
( x  =  a  ->  B  =  C )  <->  ( x  =  A  ->  B  =  C ) ) )
32albidv 1635 . 2  |-  ( a  =  A  ->  ( A. x ( x  =  a  ->  B  =  C )  <->  A. x
( x  =  A  ->  B  =  C ) ) )
4 csbeq1 3254 . . 3  |-  ( a  =  A  ->  [_ a  /  x ]_ B  = 
[_ A  /  x ]_ B )
54eqeq1d 2444 . 2  |-  ( a  =  A  ->  ( [_ a  /  x ]_ B  =  C  <->  [_ A  /  x ]_ B  =  C )
)
6 vex 2959 . . 3  |-  a  e. 
_V
7 csbiebg.2 . . 3  |-  F/_ x C
86, 7csbieb 3289 . 2  |-  ( A. x ( x  =  a  ->  B  =  C )  <->  [_ a  /  x ]_ B  =  C )
93, 5, 8vtoclbg 3012 1  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549    = wceq 1652    e. wcel 1725   F/_wnfc 2559   [_csb 3251
This theorem is referenced by:  cdlemefrs29bpre0  31193
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-sbc 3162  df-csb 3252
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