ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  r19.30dc Unicode version

Theorem r19.30dc 2604
Description: Restricted quantifier version of 19.30dc 1607. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 18-Jun-2023.)
Assertion
Ref Expression
r19.30dc  |-  ( ( A. x  e.  A  ( ph  \/  ps )  /\ DECID  E. x  e.  A  ps )  ->  ( A. x  e.  A  ph  \/  E. x  e.  A  ps ) )

Proof of Theorem r19.30dc
StepHypRef Expression
1 ralnex 2445 . . . . 5  |-  ( A. x  e.  A  -.  ps 
<->  -.  E. x  e.  A  ps )
2 pm2.53 712 . . . . . . 7  |-  ( ( ps  \/  ph )  ->  ( -.  ps  ->  ph ) )
32orcoms 720 . . . . . 6  |-  ( (
ph  \/  ps )  ->  ( -.  ps  ->  ph ) )
43ral2imi 2522 . . . . 5  |-  ( A. x  e.  A  ( ph  \/  ps )  -> 
( A. x  e.  A  -.  ps  ->  A. x  e.  A  ph ) )
51, 4syl5bir 152 . . . 4  |-  ( A. x  e.  A  ( ph  \/  ps )  -> 
( -.  E. x  e.  A  ps  ->  A. x  e.  A  ph ) )
65adantr 274 . . 3  |-  ( ( A. x  e.  A  ( ph  \/  ps )  /\ DECID  E. x  e.  A  ps )  ->  ( -.  E. x  e.  A  ps  ->  A. x  e.  A  ph ) )
7 dfordc 878 . . . 4  |-  (DECID  E. x  e.  A  ps  ->  ( ( E. x  e.  A  ps  \/  A. x  e.  A  ph )  <->  ( -.  E. x  e.  A  ps  ->  A. x  e.  A  ph ) ) )
87adantl 275 . . 3  |-  ( ( A. x  e.  A  ( ph  \/  ps )  /\ DECID  E. x  e.  A  ps )  ->  ( ( E. x  e.  A  ps  \/  A. x  e.  A  ph )  <->  ( -.  E. x  e.  A  ps  ->  A. x  e.  A  ph ) ) )
96, 8mpbird 166 . 2  |-  ( ( A. x  e.  A  ( ph  \/  ps )  /\ DECID  E. x  e.  A  ps )  ->  ( E. x  e.  A  ps  \/  A. x  e.  A  ph ) )
109orcomd 719 1  |-  ( ( A. x  e.  A  ( ph  \/  ps )  /\ DECID  E. x  e.  A  ps )  ->  ( A. x  e.  A  ph  \/  E. x  e.  A  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698  DECID wdc 820   A.wral 2435   E.wrex 2436
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-gen 1429  ax-ie2 1474
This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1338  df-fal 1341  df-ral 2440  df-rex 2441
This theorem is referenced by:  exmidontriimlem1  7157
  Copyright terms: Public domain W3C validator