Theorem alexdc 1630

Description: Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1656. (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 1552	. . 3 ⊢ Ⅎ𝑥∀𝑥DECID 𝜑
2		notnotbdc 873	. . . 4 ⊢ (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
3	2	sps 1548	. . 3 ⊢ (∀𝑥DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
4	1, 3	albid 1626	. 2 ⊢ (∀𝑥DECID 𝜑 → (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑))
5		alnex 1510	. 2 ⊢ (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
6	4, 5	bitrdi 196	1 ⊢ (∀𝑥DECID 𝜑 → (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  DECID wdc 835  ∀wal 1362  ∃wex 1503

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-gen 1460  ax-ie2 1505  ax-4 1521  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-fal 1370  df-nf 1472

This theorem is referenced by: (None)