Theorem r19.32r 2501

Description: One direction of Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. For decidable propositions this is an equivalence. (Contributed by Jim Kingdon, 19-Aug-2018.)

Hypothesis

Ref	Expression
r19.32r.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
r19.32r	⊢ ((𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓))

Proof of Theorem r19.32r

Step	Hyp	Ref	Expression
1		r19.32r.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		orc 665	. . . . 5 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
3	2	a1d 22	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜑 ∨ 𝜓)))
4	1, 3	alrimi 1455	. . 3 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ∨ 𝜓)))
5		df-ral 2353	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
6		olc 664	. . . . . 6 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
7	6	imim2i 12	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝜓) → (𝑥 ∈ 𝐴 → (𝜑 ∨ 𝜓)))
8	7	alimi 1384	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) → ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ∨ 𝜓)))
9	5, 8	sylbi 119	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ∨ 𝜓)))
10	4, 9	jaoi 668	. 2 ⊢ ((𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ∨ 𝜓)))
11		df-ral 2353	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 ∨ 𝜓)))
12	10, 11	sylibr 132	1 ⊢ ((𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓))

Syntax hints:   → wi 4   ∨ wo 661  ∀wal 1282  Ⅎwnf 1389   ∈ wcel 1433  ∀wral 2348

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-io 662  ax-5 1376  ax-gen 1378  ax-4 1440

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-ral 2353

This theorem is referenced by:  r19.32vr  2502