Theorem ralm 3513

Description: Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.)

Assertion

Ref	Expression
ralm	⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑)

Proof of Theorem ralm

Step	Hyp	Ref	Expression
1		df-ral 2449	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
2	1	imbi2i 225	. . . . 5 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)))
3		19.38 1664	. . . . 5 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝜑)))
4	2, 3	sylbi 120	. . . 4 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) → ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝜑)))
5		pm2.43 53	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝜑)) → (𝑥 ∈ 𝐴 → 𝜑))
6	5	alimi 1443	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝜑)) → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
7	4, 6	syl 14	. . 3 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
8	7, 1	sylibr 133	. 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) → ∀𝑥 ∈ 𝐴 𝜑)
9		ax-1 6	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑))
10	8, 9	impbii 125	1 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1341  ∃wex 1480   ∈ wcel 2136  ∀wral 2444

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-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-ral 2449

This theorem is referenced by:  raaan  3515