Theorem dfexdc 1489

Description: Defining E. x ph given decidability. It is common in classical logic to define E. x ph as -. A. x -. ph but in intuitionistic logic without a decidability condition, that is only an implication not an equivalence, as seen at exalim 1490. (Contributed by Jim Kingdon, 15-Mar-2018.)

Assertion

Ref	Expression
dfexdc	|- (DECID E. x ph -> ( E. x ph <-> -. A. x -. ph ) )

Proof of Theorem dfexdc

Step	Hyp	Ref	Expression
1		alnex 1487	. . 3 |- ( A. x -. ph <-> -. E. x ph )
2	1	a1i 9	. 2 |- (DECID E. x ph -> ( A. x -. ph <-> -. E. x ph ) )
3	2	con2biidc 869	1 |- (DECID E. x ph -> ( E. x ph <-> -. A. x -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104  DECID wdc 824   A.wal 1341   E.wex 1480

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 1435  ax-gen 1437  ax-ie2 1482

This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1346  df-fal 1349

This theorem is referenced by:  dfrex2dc  2457