Theorem alexdc 1667

Description: Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1693. (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 1589	. . 3 ⊢ Ⅎ𝑥∀𝑥DECID 𝜑
2		notnotbdc 879	. . . 4 ⊢ (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
3	2	sps 1585	. . 3 ⊢ (∀𝑥DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
4	1, 3	albid 1663	. 2 ⊢ (∀𝑥DECID 𝜑 → (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑))
5		alnex 1547	. 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 841  ∀wal 1395  ∃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  ax-4 1558  ax-ial 1582

This theorem depends on definitions:  df-bi 117  df-dc 842  df-tru 1400  df-fal 1403  df-nf 1509

This theorem is referenced by: (None)