Theorem bj-hbext 34039

Description: Closed form of hbex 2340. (Contributed by BJ, 10-Oct-2019.)

Assertion

Ref	Expression
bj-hbext	⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑))

Proof of Theorem bj-hbext

Step	Hyp	Ref	Expression
1		nfa2 2172	. . . 4 ⊢ Ⅎ𝑥∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑)
2		hbnt 2298	. . . . . 6 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
3	2	alimi 1808	. . . . 5 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑))
4		bj-hbalt 34010	. . . . 5 ⊢ (∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑) → (∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑))
5	3, 4	syl 17	. . . 4 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑))
6	1, 5	alrimi 2209	. . 3 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑))
7		hbnt 2298	. . 3 ⊢ (∀𝑥(∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑) → (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑))
8	6, 7	syl 17	. 2 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑))
9		df-ex 1777	. . 3 ⊢ (∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑)
10	9	bicomi 226	. 2 ⊢ (¬ ∀𝑦 ¬ 𝜑 ↔ ∃𝑦𝜑)
11	10	albii 1816	. 2 ⊢ (∀𝑥 ¬ ∀𝑦 ¬ 𝜑 ↔ ∀𝑥∃𝑦𝜑)
12	8, 10, 11	3imtr3g 297	1 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1531  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-11 2157  ax-12 2173

This theorem depends on definitions:  df-bi 209  df-or 844  df-ex 1777  df-nf 1781

This theorem is referenced by:  bj-nfext  34041