Theorem 19.30dc 1627

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	⊢ (DECID ∃𝑥𝜓 → (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓)))

Proof of Theorem 19.30dc

Step	Hyp	Ref	Expression
1		df-dc 835	. 2 ⊢ (DECID ∃𝑥𝜓 ↔ (∃𝑥𝜓 ∨ ¬ ∃𝑥𝜓))
2		olc 711	. . . 4 ⊢ (∃𝑥𝜓 → (∀𝑥𝜑 ∨ ∃𝑥𝜓))
3	2	a1d 22	. . 3 ⊢ (∃𝑥𝜓 → (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓)))
4		alnex 1499	. . . . 5 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
5		orel2 726	. . . . . 6 ⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓) → 𝜑))
6	5	al2imi 1458	. . . . 5 ⊢ (∀𝑥 ¬ 𝜓 → (∀𝑥(𝜑 ∨ 𝜓) → ∀𝑥𝜑))
7	4, 6	sylbir 135	. . . 4 ⊢ (¬ ∃𝑥𝜓 → (∀𝑥(𝜑 ∨ 𝜓) → ∀𝑥𝜑))
8		orc 712	. . . 4 ⊢ (∀𝑥𝜑 → (∀𝑥𝜑 ∨ ∃𝑥𝜓))
9	7, 8	syl6 33	. . 3 ⊢ (¬ ∃𝑥𝜓 → (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓)))
10	3, 9	jaoi 716	. 2 ⊢ ((∃𝑥𝜓 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓)))
11	1, 10	sylbi 121	1 ⊢ (DECID ∃𝑥𝜓 → (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 708  DECID wdc 834  ∀wal 1351  ∃wex 1492

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

This theorem is referenced by: (None)