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Theorem dfrex2dc 2429
 Description: Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 29-Jun-2022.)
Assertion
Ref Expression
dfrex2dc (DECID𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑))

Proof of Theorem dfrex2dc
StepHypRef Expression
1 df-rex 2423 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
21dcbii 826 . . 3 (DECID𝑥𝐴 𝜑DECID𝑥(𝑥𝐴𝜑))
3 dfexdc 1478 . . 3 (DECID𝑥(𝑥𝐴𝜑) → (∃𝑥(𝑥𝐴𝜑) ↔ ¬ ∀𝑥 ¬ (𝑥𝐴𝜑)))
42, 3sylbi 120 . 2 (DECID𝑥𝐴 𝜑 → (∃𝑥(𝑥𝐴𝜑) ↔ ¬ ∀𝑥 ¬ (𝑥𝐴𝜑)))
5 df-ral 2422 . . . 4 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥𝐴 → ¬ 𝜑))
6 imnan 680 . . . . 5 ((𝑥𝐴 → ¬ 𝜑) ↔ ¬ (𝑥𝐴𝜑))
76albii 1447 . . . 4 (∀𝑥(𝑥𝐴 → ¬ 𝜑) ↔ ∀𝑥 ¬ (𝑥𝐴𝜑))
85, 7bitri 183 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥𝐴𝜑))
98notbii 658 . 2 (¬ ∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥 ¬ (𝑥𝐴𝜑))
104, 1, 93bitr4g 222 1 (DECID𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 820  ∀wal 1330  ∃wex 1469   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-gen 1426  ax-ie2 1471 This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-fal 1338  df-ral 2422  df-rex 2423 This theorem is referenced by:  dfrex2fin  6805  exmidomniim  7021
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