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Theorem dfrex2dc 2428
Description: Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 29-Jun-2022.)
Assertion
Ref Expression
dfrex2dc  |-  (DECID  E. x  e.  A  ph  ->  ( E. x  e.  A  ph  <->  -. 
A. x  e.  A  -.  ph ) )

Proof of Theorem dfrex2dc
StepHypRef Expression
1 df-rex 2422 . . . 4  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
21dcbii 825 . . 3  |-  (DECID  E. x  e.  A  ph  <-> DECID  E. x ( x  e.  A  /\  ph ) )
3 dfexdc 1477 . . 3  |-  (DECID  E. x
( x  e.  A  /\  ph )  ->  ( E. x ( x  e.  A  /\  ph )  <->  -. 
A. x  -.  (
x  e.  A  /\  ph ) ) )
42, 3sylbi 120 . 2  |-  (DECID  E. x  e.  A  ph  ->  ( E. x ( x  e.  A  /\  ph )  <->  -. 
A. x  -.  (
x  e.  A  /\  ph ) ) )
5 df-ral 2421 . . . 4  |-  ( A. x  e.  A  -.  ph  <->  A. x ( x  e.  A  ->  -.  ph )
)
6 imnan 679 . . . . 5  |-  ( ( x  e.  A  ->  -.  ph )  <->  -.  (
x  e.  A  /\  ph ) )
76albii 1446 . . . 4  |-  ( A. x ( x  e.  A  ->  -.  ph )  <->  A. x  -.  ( x  e.  A  /\  ph ) )
85, 7bitri 183 . . 3  |-  ( A. x  e.  A  -.  ph  <->  A. x  -.  ( x  e.  A  /\  ph ) )
98notbii 657 . 2  |-  ( -. 
A. x  e.  A  -.  ph  <->  -.  A. x  -.  ( x  e.  A  /\  ph ) )
104, 1, 93bitr4g 222 1  |-  (DECID  E. x  e.  A  ph  ->  ( E. x  e.  A  ph  <->  -. 
A. x  e.  A  -.  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104  DECID wdc 819   A.wal 1329   E.wex 1468    e. wcel 1480   A.wral 2416   E.wrex 2417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-gen 1425  ax-ie2 1470
This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-fal 1337  df-ral 2421  df-rex 2422
This theorem is referenced by:  dfrex2fin  6800  exmidomniim  7016
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