Theorem alexdc 1579

Description: Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1605. (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 1502	. . 3 ⊢ Ⅎ𝑥∀𝑥DECID 𝜑
2		notnotbdc 838	. . . 4 ⊢ (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
3	2	sps 1498	. . 3 ⊢ (∀𝑥DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
4	1, 3	albid 1575	. 2 ⊢ (∀𝑥DECID 𝜑 → (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑))
5		alnex 1456	. 2 ⊢ (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
6	4, 5	syl6bb 195	1 ⊢ (∀𝑥DECID 𝜑 → (∀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  ax-4 1468  ax-ial 1495

This theorem depends on definitions:  df-bi 116  df-dc 803  df-tru 1315  df-fal 1318  df-nf 1418

This theorem is referenced by: (None)