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Theorem alexeq 3207
Description: Two ways to express substitution of  A for  x in  ph. Obsoleted by alexeqg 3206. (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 2444 . . . . 5  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
32anbi1d 709 . . . 4  |-  ( y  =  A  ->  (
( x  =  y  /\  ph )  <->  ( x  =  A  /\  ph )
) )
43exbidv 1761 . . 3  |-  ( y  =  A  ->  ( E. x ( x  =  y  /\  ph )  <->  E. x ( x  =  A  /\  ph )
) )
52imbi1d 318 . . . 4  |-  ( y  =  A  ->  (
( x  =  y  ->  ph )  <->  ( x  =  A  ->  ph )
) )
65albidv 1760 . . 3  |-  ( y  =  A  ->  ( A. x ( x  =  y  ->  ph )  <->  A. x
( x  =  A  ->  ph ) ) )
7 sb56 2224 . . 3  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
81, 4, 6, 7vtoclb 3142 . 2  |-  ( E. x ( x  =  A  /\  ph )  <->  A. x ( x  =  A  ->  ph ) )
98bicomi 205 1  |-  ( A. x ( x  =  A  ->  ph )  <->  E. x
( x  =  A  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437   E.wex 1659    e. wcel 1870   _Vcvv 3087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-v 3089
This theorem is referenced by: (None)
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