Theorem dfexdc 1549

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 1550. (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 1547	. . 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 886	1 |- (DECID E. x ph -> ( E. x ph <-> -. A. x -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105  DECID wdc 841   A.wal 1395   E.wex 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-gen 1497  ax-ie2 1542

This theorem depends on definitions:  df-bi 117  df-dc 842  df-tru 1400  df-fal 1403

This theorem is referenced by:  dfrex2dc  2523