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Theorem bj-issetwt 30999
Description: Closed form of bj-issetw 31000. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-issetwt  |-  ( A. x ph  ->  ( A  e.  { x  |  ph } 
<->  E. y  y  =  A ) )
Distinct variable group:    y, A
Allowed substitution hints:    ph( x, y)    A( x)

Proof of Theorem bj-issetwt
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-clel 2397 . . 3  |-  ( A  e.  { x  | 
ph }  <->  E. z
( z  =  A  /\  z  e.  {
x  |  ph }
) )
21a1i 11 . 2  |-  ( A. x ph  ->  ( A  e.  { x  |  ph } 
<->  E. z ( z  =  A  /\  z  e.  { x  |  ph } ) ) )
3 bj-vexwvt 30996 . . . . 5  |-  ( A. x ph  ->  z  e.  { x  |  ph }
)
43biantrud 505 . . . 4  |-  ( A. x ph  ->  ( z  =  A  <->  ( z  =  A  /\  z  e. 
{ x  |  ph } ) ) )
54bicomd 201 . . 3  |-  ( A. x ph  ->  ( (
z  =  A  /\  z  e.  { x  |  ph } )  <->  z  =  A ) )
65exbidv 1735 . 2  |-  ( A. x ph  ->  ( E. z ( z  =  A  /\  z  e. 
{ x  |  ph } )  <->  E. z 
z  =  A ) )
7 bj-denotes 30998 . . 3  |-  ( E. z  z  =  A  <->  E. y  y  =  A )
87a1i 11 . 2  |-  ( A. x ph  ->  ( E. z  z  =  A  <->  E. y  y  =  A ) )
92, 6, 83bitrd 279 1  |-  ( A. x ph  ->  ( A  e.  { x  |  ph } 
<->  E. y  y  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1403    = wceq 1405   E.wex 1633    e. wcel 1842   {cab 2387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-12 1878
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1634  df-sb 1764  df-clab 2388  df-clel 2397
This theorem is referenced by:  bj-issetw  31000
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