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Theorem dfrex2dc 2481
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 2474 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
21dcbii 841 . . 3 (DECID𝑥𝐴 𝜑DECID𝑥(𝑥𝐴𝜑))
3 dfexdc 1512 . . 3 (DECID𝑥(𝑥𝐴𝜑) → (∃𝑥(𝑥𝐴𝜑) ↔ ¬ ∀𝑥 ¬ (𝑥𝐴𝜑)))
42, 3sylbi 121 . 2 (DECID𝑥𝐴 𝜑 → (∃𝑥(𝑥𝐴𝜑) ↔ ¬ ∀𝑥 ¬ (𝑥𝐴𝜑)))
5 df-ral 2473 . . . 4 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥𝐴 → ¬ 𝜑))
6 imnan 691 . . . . 5 ((𝑥𝐴 → ¬ 𝜑) ↔ ¬ (𝑥𝐴𝜑))
76albii 1481 . . . 4 (∀𝑥(𝑥𝐴 → ¬ 𝜑) ↔ ∀𝑥 ¬ (𝑥𝐴𝜑))
85, 7bitri 184 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥𝐴𝜑))
98notbii 669 . 2 (¬ ∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥 ¬ (𝑥𝐴𝜑))
104, 1, 93bitr4g 223 1 (DECID𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  DECID wdc 835  wal 1362  wex 1503  wcel 2160  wral 2468  wrex 2469
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-gen 1460  ax-ie2 1505
This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-fal 1370  df-ral 2473  df-rex 2474
This theorem is referenced by:  dfrex2fin  6930  exmidomniim  7168
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