Theorem alexdc 1550

Description: Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1576. (Contributed by Jim Kingdon, 2-Jun-2018.)

Assertion

Ref	Expression
alexdc	⊢ (∀𝑥DECID 𝜑 → (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑))

Proof of Theorem alexdc

Step	Hyp	Ref	Expression
1		nfa1 1474	. . 3 ⊢ Ⅎ𝑥∀𝑥DECID 𝜑
2		notnotbdc 799	. . . 4 ⊢ (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
3	2	sps 1470	. . 3 ⊢ (∀𝑥DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
4	1, 3	albid 1546	. 2 ⊢ (∀𝑥DECID 𝜑 → (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑))
5		alnex 1428	. 2 ⊢ (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
6	4, 5	syl6bb 194	1 ⊢ (∀𝑥DECID 𝜑 → (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103  DECID wdc 775  ∀wal 1282  ∃wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-gen 1378  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-dc 776  df-tru 1287  df-fal 1290  df-nf 1390

This theorem is referenced by: (None)