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Theorem 19.30dc 1563
Description: Theorem 19.30 of [Margaris] p. 90, with an additional decidability condition. (Contributed by Jim Kingdon, 21-Jul-2018.)
Assertion
Ref Expression
19.30dc (DECID𝑥𝜓 → (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓)))

Proof of Theorem 19.30dc
StepHypRef Expression
1 df-dc 781 . 2 (DECID𝑥𝜓 ↔ (∃𝑥𝜓 ∨ ¬ ∃𝑥𝜓))
2 olc 667 . . . 4 (∃𝑥𝜓 → (∀𝑥𝜑 ∨ ∃𝑥𝜓))
32a1d 22 . . 3 (∃𝑥𝜓 → (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓)))
4 alnex 1433 . . . . 5 (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
5 orel2 680 . . . . . 6 𝜓 → ((𝜑𝜓) → 𝜑))
65al2imi 1392 . . . . 5 (∀𝑥 ¬ 𝜓 → (∀𝑥(𝜑𝜓) → ∀𝑥𝜑))
74, 6sylbir 133 . . . 4 (¬ ∃𝑥𝜓 → (∀𝑥(𝜑𝜓) → ∀𝑥𝜑))
8 orc 668 . . . 4 (∀𝑥𝜑 → (∀𝑥𝜑 ∨ ∃𝑥𝜓))
97, 8syl6 33 . . 3 (¬ ∃𝑥𝜓 → (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓)))
103, 9jaoi 671 . 2 ((∃𝑥𝜓 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓)))
111, 10sylbi 119 1 (DECID𝑥𝜓 → (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 664  DECID wdc 780  wal 1287  wex 1426
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-gen 1383  ax-ie2 1428
This theorem depends on definitions:  df-bi 115  df-dc 781  df-tru 1292  df-fal 1295
This theorem is referenced by: (None)
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