Theorem ralm 3325

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 2311	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
2	1	imbi2i 215	. . . . 5 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)))
3		19.38 1566	. . . . 5 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝜑)))
4	2, 3	sylbi 114	. . . 4 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) → ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝜑)))
5		pm2.43 47	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝜑)) → (𝑥 ∈ 𝐴 → 𝜑))
6	5	alimi 1344	. . . 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 137	. 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) → ∀𝑥 ∈ 𝐴 𝜑)
9		ax-1 5	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑))
10	8, 9	impbii 117	1 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  ∃wex 1381   ∈ wcel 1393  ∀wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-ral 2311

This theorem is referenced by:  raaan  3327