Theorem 19.21ht 1569

Description: Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (New usage is discouraged.)

Assertion

Ref	Expression
19.21ht	⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Proof of Theorem 19.21ht

Step	Hyp	Ref	Expression
1		alim 1445	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
2	1	imim2d 54	. . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜓)))
3	2	com12 30	. . 3 ⊢ ((𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
4	3	sps 1525	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
5		hba1 1528	. . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑥(𝜑 → ∀𝑥𝜑))
6		ax-4 1498	. . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑))
7		hba1 1528	. . . . 5 ⊢ (∀𝑥𝜓 → ∀𝑥∀𝑥𝜓)
8	7	a1i 9	. . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥𝜓 → ∀𝑥∀𝑥𝜓))
9	5, 6, 8	hbimd 1561	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓)))
10		ax-4 1498	. . . . 5 ⊢ (∀𝑥𝜓 → 𝜓)
11	10	imim2i 12	. . . 4 ⊢ ((𝜑 → ∀𝑥𝜓) → (𝜑 → 𝜓))
12	11	alimi 1443	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
13	9, 12	syl6 33	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)))
14	4, 13	impbid 128	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-4 1498  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  19.21t  1570  sbal2  2008