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

Theorem alexeq 3194
Description: Two ways to express substitution of  A for  x in  ph. Obsoleted by alexeqg 3193. (Contributed by NM, 2-Mar-1995.) Obsolete as of 1-May-2019. (New usage is discouraged.)
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 2469 . . . . 5  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
32anbi1d 704 . . . 4  |-  ( y  =  A  ->  (
( x  =  y  /\  ph )  <->  ( x  =  A  /\  ph )
) )
43exbidv 1681 . . 3  |-  ( y  =  A  ->  ( E. x ( x  =  y  /\  ph )  <->  E. x ( x  =  A  /\  ph )
) )
52imbi1d 317 . . . 4  |-  ( y  =  A  ->  (
( x  =  y  ->  ph )  <->  ( x  =  A  ->  ph )
) )
65albidv 1680 . . 3  |-  ( y  =  A  ->  ( A. x ( x  =  y  ->  ph )  <->  A. x
( x  =  A  ->  ph ) ) )
7 sb56 2141 . . 3  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
81, 4, 6, 7vtoclb 3131 . 2  |-  ( E. x ( x  =  A  /\  ph )  <->  A. x ( x  =  A  ->  ph ) )
98bicomi 202 1  |-  ( A. x ( x  =  A  ->  ph )  <->  E. x
( x  =  A  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368    = wceq 1370   E.wex 1587    e. wcel 1758   _Vcvv 3076
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-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3078
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator