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Theorem dfrex2dc 2365
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 2359 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
21dcbii 781 . . 3 (DECID𝑥𝐴 𝜑DECID𝑥(𝑥𝐴𝜑))
3 dfexdc 1431 . . 3 (DECID𝑥(𝑥𝐴𝜑) → (∃𝑥(𝑥𝐴𝜑) ↔ ¬ ∀𝑥 ¬ (𝑥𝐴𝜑)))
42, 3sylbi 119 . 2 (DECID𝑥𝐴 𝜑 → (∃𝑥(𝑥𝐴𝜑) ↔ ¬ ∀𝑥 ¬ (𝑥𝐴𝜑)))
5 df-ral 2358 . . . 4 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥𝐴 → ¬ 𝜑))
6 imnan 657 . . . . 5 ((𝑥𝐴 → ¬ 𝜑) ↔ ¬ (𝑥𝐴𝜑))
76albii 1400 . . . 4 (∀𝑥(𝑥𝐴 → ¬ 𝜑) ↔ ∀𝑥 ¬ (𝑥𝐴𝜑))
85, 7bitri 182 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥𝐴𝜑))
98notbii 627 . 2 (¬ ∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥 ¬ (𝑥𝐴𝜑))
104, 1, 93bitr4g 221 1 (DECID𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  DECID wdc 776  wal 1283  wex 1422  wcel 1434  wral 2353  wrex 2354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-gen 1379  ax-ie2 1424
This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-fal 1291  df-ral 2358  df-rex 2359
This theorem is referenced by:  dfrex2fin  6544  exmidomniim  6700
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