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Theorem r19.30dc 2624
Description: Restricted quantifier version of 19.30dc 1627. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 18-Jun-2023.)
Assertion
Ref Expression
r19.30dc ((∀𝑥𝐴 (𝜑𝜓) ∧ DECID𝑥𝐴 𝜓) → (∀𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))

Proof of Theorem r19.30dc
StepHypRef Expression
1 ralnex 2465 . . . . 5 (∀𝑥𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥𝐴 𝜓)
2 pm2.53 722 . . . . . . 7 ((𝜓𝜑) → (¬ 𝜓𝜑))
32orcoms 730 . . . . . 6 ((𝜑𝜓) → (¬ 𝜓𝜑))
43ral2imi 2542 . . . . 5 (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 ¬ 𝜓 → ∀𝑥𝐴 𝜑))
51, 4biimtrrid 153 . . . 4 (∀𝑥𝐴 (𝜑𝜓) → (¬ ∃𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜑))
65adantr 276 . . 3 ((∀𝑥𝐴 (𝜑𝜓) ∧ DECID𝑥𝐴 𝜓) → (¬ ∃𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜑))
7 dfordc 892 . . . 4 (DECID𝑥𝐴 𝜓 → ((∃𝑥𝐴 𝜓 ∨ ∀𝑥𝐴 𝜑) ↔ (¬ ∃𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜑)))
87adantl 277 . . 3 ((∀𝑥𝐴 (𝜑𝜓) ∧ DECID𝑥𝐴 𝜓) → ((∃𝑥𝐴 𝜓 ∨ ∀𝑥𝐴 𝜑) ↔ (¬ ∃𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜑)))
96, 8mpbird 167 . 2 ((∀𝑥𝐴 (𝜑𝜓) ∧ DECID𝑥𝐴 𝜓) → (∃𝑥𝐴 𝜓 ∨ ∀𝑥𝐴 𝜑))
109orcomd 729 1 ((∀𝑥𝐴 (𝜑𝜓) ∧ DECID𝑥𝐴 𝜓) → (∀𝑥𝐴 𝜑 ∨ ∃𝑥𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834  wral 2455  wrex 2456
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-gen 1449  ax-ie2 1494
This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-fal 1359  df-ral 2460  df-rex 2461
This theorem is referenced by:  exmidontriimlem1  7219
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