Theorem bj-nnfext 34115

Description: See nfex 2342 and bj-nfext 34065. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem bj-nnfext

Step	Hyp	Ref	Expression
1		df-bj-nnf 34075	. . . 4 ⊢ (Ⅎ'𝑦𝜑 ↔ ((∃𝑦𝜑 → 𝜑) ∧ (𝜑 → ∀𝑦𝜑)))
2	1	albii 1819	. . 3 ⊢ (∀𝑥Ⅎ'𝑦𝜑 ↔ ∀𝑥((∃𝑦𝜑 → 𝜑) ∧ (𝜑 → ∀𝑦𝜑)))
3		simpl 485	. . . . 5 ⊢ (((∃𝑦𝜑 → 𝜑) ∧ (𝜑 → ∀𝑦𝜑)) → (∃𝑦𝜑 → 𝜑))
4	3	alimi 1811	. . . 4 ⊢ (∀𝑥((∃𝑦𝜑 → 𝜑) ∧ (𝜑 → ∀𝑦𝜑)) → ∀𝑥(∃𝑦𝜑 → 𝜑))
5		bj-nnflemee 34111	. . . 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 219	. 2 ⊢ (∀𝑥Ⅎ'𝑦𝜑 → (∃𝑦∃𝑥𝜑 → ∃𝑥𝜑))
8		simpr 487	. . . . 5 ⊢ (((∃𝑦𝜑 → 𝜑) ∧ (𝜑 → ∀𝑦𝜑)) → (𝜑 → ∀𝑦𝜑))
9	8	alimi 1811	. . . 4 ⊢ (∀𝑥((∃𝑦𝜑 → 𝜑) ∧ (𝜑 → ∀𝑦𝜑)) → ∀𝑥(𝜑 → ∀𝑦𝜑))
10		bj-nnflemae 34112	. . . 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 219	. 2 ⊢ (∀𝑥Ⅎ'𝑦𝜑 → (∃𝑥𝜑 → ∀𝑦∃𝑥𝜑))
13		df-bj-nnf 34075	. 2 ⊢ (Ⅎ'𝑦∃𝑥𝜑 ↔ ((∃𝑦∃𝑥𝜑 → ∃𝑥𝜑) ∧ (∃𝑥𝜑 → ∀𝑦∃𝑥𝜑)))
14	7, 12, 13	sylanbrc 585	1 ⊢ (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦∃𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1534  ∃wex 1779  Ⅎ'wnnf 34074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-11 2160  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-bj-nnf 34075

This theorem is referenced by: (None)