Theorem bj-nnfalt 34875

Description: See nfal 2321 and bj-nfalt 34820. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-nnfalt	⊢ (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦∀𝑥𝜑)

Proof of Theorem bj-nnfalt

Step	Hyp	Ref	Expression
1		df-bj-nnf 34833	. . . 4 ⊢ (Ⅎ'𝑦𝜑 ↔ ((∃𝑦𝜑 → 𝜑) ∧ (𝜑 → ∀𝑦𝜑)))
2	1	albii 1823	. . 3 ⊢ (∀𝑥Ⅎ'𝑦𝜑 ↔ ∀𝑥((∃𝑦𝜑 → 𝜑) ∧ (𝜑 → ∀𝑦𝜑)))
3		simpl 482	. . . . 5 ⊢ (((∃𝑦𝜑 → 𝜑) ∧ (𝜑 → ∀𝑦𝜑)) → (∃𝑦𝜑 → 𝜑))
4	3	alimi 1815	. . . 4 ⊢ (∀𝑥((∃𝑦𝜑 → 𝜑) ∧ (𝜑 → ∀𝑦𝜑)) → ∀𝑥(∃𝑦𝜑 → 𝜑))
5		bj-nnflemea 34874	. . . 4 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜑) → (∃𝑦∀𝑥𝜑 → ∀𝑥𝜑))
6	4, 5	syl 17	. . 3 ⊢ (∀𝑥((∃𝑦𝜑 → 𝜑) ∧ (𝜑 → ∀𝑦𝜑)) → (∃𝑦∀𝑥𝜑 → ∀𝑥𝜑))
7	2, 6	sylbi 216	. 2 ⊢ (∀𝑥Ⅎ'𝑦𝜑 → (∃𝑦∀𝑥𝜑 → ∀𝑥𝜑))
8		simpr 484	. . . . 5 ⊢ (((∃𝑦𝜑 → 𝜑) ∧ (𝜑 → ∀𝑦𝜑)) → (𝜑 → ∀𝑦𝜑))
9	8	alimi 1815	. . . 4 ⊢ (∀𝑥((∃𝑦𝜑 → 𝜑) ∧ (𝜑 → ∀𝑦𝜑)) → ∀𝑥(𝜑 → ∀𝑦𝜑))
10		bj-nnflemaa 34871	. . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑))
11	9, 10	syl 17	. . 3 ⊢ (∀𝑥((∃𝑦𝜑 → 𝜑) ∧ (𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑))
12	2, 11	sylbi 216	. 2 ⊢ (∀𝑥Ⅎ'𝑦𝜑 → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑))
13		df-bj-nnf 34833	. 2 ⊢ (Ⅎ'𝑦∀𝑥𝜑 ↔ ((∃𝑦∀𝑥𝜑 → ∀𝑥𝜑) ∧ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)))
14	7, 12, 13	sylanbrc 582	1 ⊢ (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦∀𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537  ∃wex 1783  Ⅎ'wnnf 34832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-bj-nnf 34833

This theorem is referenced by:  bj-pm11.53v  34886