Theorem r19.30dc 2617

Description: Restricted quantifier version of 19.30dc 1620. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 18-Jun-2023.)

Assertion

Ref	Expression
r19.30dc	⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ∧ DECID ∃𝑥 ∈ 𝐴 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))

Proof of Theorem r19.30dc

Step	Hyp	Ref	Expression
1		ralnex 2458	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜓)
2		pm2.53 717	. . . . . . 7 ⊢ ((𝜓 ∨ 𝜑) → (¬ 𝜓 → 𝜑))
3	2	orcoms 725	. . . . . 6 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜓 → 𝜑))
4	3	ral2imi 2535	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑))
5	1, 4	syl5bir 152	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (¬ ∃𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜑))
6	5	adantr 274	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ∧ DECID ∃𝑥 ∈ 𝐴 𝜓) → (¬ ∃𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜑))
7		dfordc 887	. . . 4 ⊢ (DECID ∃𝑥 ∈ 𝐴 𝜓 → ((∃𝑥 ∈ 𝐴 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜑) ↔ (¬ ∃𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜑)))
8	7	adantl 275	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ∧ DECID ∃𝑥 ∈ 𝐴 𝜓) → ((∃𝑥 ∈ 𝐴 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜑) ↔ (¬ ∃𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜑)))
9	6, 8	mpbird 166	. 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ∧ DECID ∃𝑥 ∈ 𝐴 𝜓) → (∃𝑥 ∈ 𝐴 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜑))
10	9	orcomd 724	1 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ∧ DECID ∃𝑥 ∈ 𝐴 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703  DECID wdc 829  ∀wral 2448  ∃wrex 2449

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-gen 1442  ax-ie2 1487

This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-fal 1354  df-ral 2453  df-rex 2454

This theorem is referenced by:  exmidontriimlem1  7198