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Theorem dfrex2dc 2461
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 2454 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
21dcbii 835 . . 3 (DECID𝑥𝐴 𝜑DECID𝑥(𝑥𝐴𝜑))
3 dfexdc 1494 . . 3 (DECID𝑥(𝑥𝐴𝜑) → (∃𝑥(𝑥𝐴𝜑) ↔ ¬ ∀𝑥 ¬ (𝑥𝐴𝜑)))
42, 3sylbi 120 . 2 (DECID𝑥𝐴 𝜑 → (∃𝑥(𝑥𝐴𝜑) ↔ ¬ ∀𝑥 ¬ (𝑥𝐴𝜑)))
5 df-ral 2453 . . . 4 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥𝐴 → ¬ 𝜑))
6 imnan 685 . . . . 5 ((𝑥𝐴 → ¬ 𝜑) ↔ ¬ (𝑥𝐴𝜑))
76albii 1463 . . . 4 (∀𝑥(𝑥𝐴 → ¬ 𝜑) ↔ ∀𝑥 ¬ (𝑥𝐴𝜑))
85, 7bitri 183 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥𝐴𝜑))
98notbii 663 . 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 829  wal 1346  wex 1485  wcel 2141  wral 2448  wrex 2449
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-gen 1442  ax-ie2 1487
This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-fal 1354  df-ral 2453  df-rex 2454
This theorem is referenced by:  dfrex2fin  6877  exmidomniim  7113
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