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Theorem alexdc 1526
Description: Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1552. (Contributed by Jim Kingdon, 2-Jun-2018.)
Assertion
Ref Expression
alexdc (∀𝑥DECID 𝜑 → (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑))

Proof of Theorem alexdc
StepHypRef Expression
1 nfa1 1450 . . 3 𝑥𝑥DECID 𝜑
2 notnotbdc 777 . . . 4 (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
32sps 1446 . . 3 (∀𝑥DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑))
41, 3albid 1522 . 2 (∀𝑥DECID 𝜑 → (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑))
5 alnex 1404 . 2 (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
64, 5syl6bb 189 1 (∀𝑥DECID 𝜑 → (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102  DECID wdc 753  wal 1257  wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-gen 1354  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-dc 754  df-tru 1262  df-fal 1265  df-nf 1366
This theorem is referenced by: (None)
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