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

Theorem alexeq 3067
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 2447 . . . . 5  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
32anbi1d 687 . . . 4  |-  ( y  =  A  ->  (
( x  =  y  /\  ph )  <->  ( x  =  A  /\  ph )
) )
43exbidv 1637 . . 3  |-  ( y  =  A  ->  ( E. x ( x  =  y  /\  ph )  <->  E. x ( x  =  A  /\  ph )
) )
52imbi1d 310 . . . 4  |-  ( y  =  A  ->  (
( x  =  y  ->  ph )  <->  ( x  =  A  ->  ph )
) )
65albidv 1636 . . 3  |-  ( y  =  A  ->  ( A. x ( x  =  y  ->  ph )  <->  A. x
( x  =  A  ->  ph ) ) )
7 sb56 2176 . . 3  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
81, 4, 6, 7vtoclb 3011 . 2  |-  ( E. x ( x  =  A  /\  ph )  <->  A. x ( x  =  A  ->  ph ) )
98bicomi 195 1  |-  ( A. x ( x  =  A  ->  ph )  <->  E. x
( x  =  A  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1726   _Vcvv 2958
This theorem is referenced by:  ceqex  3068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-v 2960
  Copyright terms: Public domain W3C validator