Theorem dfexdc 1458

Description: Defining ∃𝑥𝜑 given decidability. It is common in classical logic to define ∃𝑥𝜑 as ¬ ∀𝑥¬ 𝜑 but in intuitionistic logic without a decidability condition, that is only an implication not an equivalence, as seen at exalim 1459. (Contributed by Jim Kingdon, 15-Mar-2018.)

Assertion

Ref	Expression
dfexdc	⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑))

Proof of Theorem dfexdc

Step	Hyp	Ref	Expression
1		alnex 1456	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2	1	a1i 9	. 2 ⊢ (DECID ∃𝑥𝜑 → (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑))
3	2	con2biidc 845	1 ⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  DECID wdc 802  ∀wal 1310  ∃wex 1449

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-gen 1406  ax-ie2 1451

This theorem depends on definitions:  df-bi 116  df-dc 803  df-tru 1315  df-fal 1318

This theorem is referenced by:  dfrex2dc  2400