ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  alexeq Unicode version

Theorem alexeq 2814
Description: Two ways to express substitution of  A for  x in  ph. (Contributed by NM, 2-Mar-1995.)
Hypothesis
Ref Expression
alexeq.1  |-  A  e. 
_V
Assertion
Ref Expression
alexeq  |-  ( A. x ( x  =  A  ->  ph )  <->  E. x
( x  =  A  /\  ph ) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem alexeq
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 alexeq.1 . . 3  |-  A  e. 
_V
2 eqeq2 2150 . . . . 5  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
32anbi1d 461 . . . 4  |-  ( y  =  A  ->  (
( x  =  y  /\  ph )  <->  ( x  =  A  /\  ph )
) )
43exbidv 1798 . . 3  |-  ( y  =  A  ->  ( E. x ( x  =  y  /\  ph )  <->  E. x ( x  =  A  /\  ph )
) )
52imbi1d 230 . . . 4  |-  ( y  =  A  ->  (
( x  =  y  ->  ph )  <->  ( x  =  A  ->  ph )
) )
65albidv 1797 . . 3  |-  ( y  =  A  ->  ( A. x ( x  =  y  ->  ph )  <->  A. x
( x  =  A  ->  ph ) ) )
7 sb56 1858 . . 3  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
81, 4, 6, 7vtoclb 2746 . 2  |-  ( E. x ( x  =  A  /\  ph )  <->  A. x ( x  =  A  ->  ph ) )
98bicomi 131 1  |-  ( A. x ( x  =  A  ->  ph )  <->  E. x
( x  =  A  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1330    = wceq 1332   E.wex 1469    e. wcel 1481   _Vcvv 2689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2691
This theorem is referenced by:  ceqex  2815
  Copyright terms: Public domain W3C validator