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

Theorem alexeq 2910
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 2305 . . . . 5  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
32anbi1d 685 . . . 4  |-  ( y  =  A  ->  (
( x  =  y  /\  ph )  <->  ( x  =  A  /\  ph )
) )
43exbidv 1616 . . 3  |-  ( y  =  A  ->  ( E. x ( x  =  y  /\  ph )  <->  E. x ( x  =  A  /\  ph )
) )
52imbi1d 308 . . . 4  |-  ( y  =  A  ->  (
( x  =  y  ->  ph )  <->  ( x  =  A  ->  ph )
) )
65albidv 1615 . . 3  |-  ( y  =  A  ->  ( A. x ( x  =  y  ->  ph )  <->  A. x
( x  =  A  ->  ph ) ) )
7 sb56 2050 . . 3  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
81, 4, 6, 7vtoclb 2854 . 2  |-  ( E. x ( x  =  A  /\  ph )  <->  A. x ( x  =  A  ->  ph ) )
98bicomi 193 1  |-  ( A. x ( x  =  A  ->  ph )  <->  E. x
( x  =  A  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801
This theorem is referenced by:  ceqex  2911  fates  25058  alexeqd  25065
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803
  Copyright terms: Public domain W3C validator