MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbiebg Unicode version

Theorem csbiebg 3120
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 2292 . . . 4  |-  ( a  =  A  ->  (
x  =  a  <->  x  =  A ) )
21imbi1d 308 . . 3  |-  ( a  =  A  ->  (
( x  =  a  ->  B  =  C )  <->  ( x  =  A  ->  B  =  C ) ) )
32albidv 1611 . 2  |-  ( a  =  A  ->  ( A. x ( x  =  a  ->  B  =  C )  <->  A. x
( x  =  A  ->  B  =  C ) ) )
4 csbeq1 3084 . . 3  |-  ( a  =  A  ->  [_ a  /  x ]_ B  = 
[_ A  /  x ]_ B )
54eqeq1d 2291 . 2  |-  ( a  =  A  ->  ( [_ a  /  x ]_ B  =  C  <->  [_ A  /  x ]_ B  =  C )
)
6 vex 2791 . . 3  |-  a  e. 
_V
7 csbiebg.2 . . 3  |-  F/_ x C
86, 7csbieb 3119 . 2  |-  ( A. x ( x  =  a  ->  B  =  C )  <->  [_ a  /  x ]_ B  =  C )
93, 5, 8vtoclbg 2844 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 176   A.wal 1527    = wceq 1623    e. wcel 1684   F/_wnfc 2406   [_csb 3081
This theorem is referenced by:  cdlemefrs29bpre0  30585
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992  df-csb 3082
  Copyright terms: Public domain W3C validator