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Theorem csbiebg 3443
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 2469 . . . 4  |-  ( a  =  A  ->  (
x  =  a  <->  x  =  A ) )
21imbi1d 315 . . 3  |-  ( a  =  A  ->  (
( x  =  a  ->  B  =  C )  <->  ( x  =  A  ->  B  =  C ) ) )
32albidv 1718 . 2  |-  ( a  =  A  ->  ( A. x ( x  =  a  ->  B  =  C )  <->  A. x
( x  =  A  ->  B  =  C ) ) )
4 csbeq1 3423 . . 3  |-  ( a  =  A  ->  [_ a  /  x ]_ B  = 
[_ A  /  x ]_ B )
54eqeq1d 2456 . 2  |-  ( a  =  A  ->  ( [_ a  /  x ]_ B  =  C  <->  [_ A  /  x ]_ B  =  C )
)
6 vex 3109 . . 3  |-  a  e. 
_V
7 csbiebg.2 . . 3  |-  F/_ x C
86, 7csbieb 3442 . 2  |-  ( A. x ( x  =  a  ->  B  =  C )  <->  [_ a  /  x ]_ B  =  C )
93, 5, 8vtoclbg 3165 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 1396    = wceq 1398    e. wcel 1823   F/_wnfc 2602   [_csb 3420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-sbc 3325  df-csb 3421
This theorem is referenced by:  cdlemefrs29bpre0  36538
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