Theorem r19.30dc 2678

Description: Restricted quantifier version of 19.30dc 1673. (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 2518	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜓)
2		pm2.53 727	. . . . . . 7 ⊢ ((𝜓 ∨ 𝜑) → (¬ 𝜓 → 𝜑))
3	2	orcoms 735	. . . . . 6 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜓 → 𝜑))
4	3	ral2imi 2595	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∀𝑥 ∈ 𝐴 ¬ 𝜓 → ∀𝑥 ∈ 𝐴 𝜑))
5	1, 4	biimtrrid 153	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (¬ ∃𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜑))
6	5	adantr 276	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ∧ DECID ∃𝑥 ∈ 𝐴 𝜓) → (¬ ∃𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜑))
7		dfordc 897	. . . 4 ⊢ (DECID ∃𝑥 ∈ 𝐴 𝜓 → ((∃𝑥 ∈ 𝐴 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜑) ↔ (¬ ∃𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜑)))
8	7	adantl 277	. . 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 167	. 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ∧ DECID ∃𝑥 ∈ 𝐴 𝜓) → (∃𝑥 ∈ 𝐴 𝜓 ∨ ∀𝑥 ∈ 𝐴 𝜑))
10	9	orcomd 734	1 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ∧ DECID ∃𝑥 ∈ 𝐴 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713  DECID wdc 839  ∀wral 2508  ∃wrex 2509

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-gen 1495  ax-ie2 1540

This theorem depends on definitions:  df-bi 117  df-dc 840  df-tru 1398  df-fal 1401  df-ral 2513  df-rex 2514

This theorem is referenced by:  exmidontriimlem1  7399