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Theorem 19.30dc 1589
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 803 . 2 (DECID𝑥𝜓 ↔ (∃𝑥𝜓 ∨ ¬ ∃𝑥𝜓))
2 olc 683 . . . 4 (∃𝑥𝜓 → (∀𝑥𝜑 ∨ ∃𝑥𝜓))
32a1d 22 . . 3 (∃𝑥𝜓 → (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓)))
4 alnex 1458 . . . . 5 (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
5 orel2 698 . . . . . 6 𝜓 → ((𝜑𝜓) → 𝜑))
65al2imi 1417 . . . . 5 (∀𝑥 ¬ 𝜓 → (∀𝑥(𝜑𝜓) → ∀𝑥𝜑))
74, 6sylbir 134 . . . 4 (¬ ∃𝑥𝜓 → (∀𝑥(𝜑𝜓) → ∀𝑥𝜑))
8 orc 684 . . . 4 (∀𝑥𝜑 → (∀𝑥𝜑 ∨ ∃𝑥𝜓))
97, 8syl6 33 . . 3 (¬ ∃𝑥𝜓 → (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓)))
103, 9jaoi 688 . 2 ((∃𝑥𝜓 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓)))
111, 10sylbi 120 1 (DECID𝑥𝜓 → (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 680  DECID wdc 802  wal 1312  wex 1451
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-gen 1408  ax-ie2 1453
This theorem depends on definitions:  df-bi 116  df-dc 803  df-tru 1317  df-fal 1320
This theorem is referenced by: (None)
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