Theorem bj-nfimt 33570

Description: Closed form of nfim 1879 and curried (exported) form of nfimt 1878. (Contributed by BJ, 20-Oct-2021.)

Assertion

Ref	Expression
bj-nfimt	⊢ (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑 → 𝜓)))

Proof of Theorem bj-nfimt

Step	Hyp	Ref	Expression
1		19.35 1860	. . . 4 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2		id 22	. . . . . 6 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
3	2	nfrd 1774	. . . . 5 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
4	3	imim1d 82	. . . 4 ⊢ (Ⅎ𝑥𝜑 → ((∀𝑥𝜑 → ∃𝑥𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓)))
5	1, 4	syl5bi 243	. . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓)))
6		id 22	. . . . . 6 ⊢ (Ⅎ𝑥𝜓 → Ⅎ𝑥𝜓)
7	6	nfrd 1774	. . . . 5 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → ∀𝑥𝜓))
8	7	imim2d 57	. . . 4 ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∃𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
9		19.38 1821	. . . 4 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
10	8, 9	syl6 35	. . 3 ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∃𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)))
11	5, 10	syl9 77	. 2 ⊢ (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → (∃𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))))
12		df-nf 1767	. 2 ⊢ (Ⅎ𝑥(𝜑 → 𝜓) ↔ (∃𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)))
13	11, 12	syl6ibr 253	1 ⊢ (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑 → 𝜓)))

Syntax hints:   → wi 4  ∀wal 1520  ∃wex 1762  Ⅎwnf 1766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792

This theorem depends on definitions:  df-bi 208  df-ex 1763  df-nf 1767

This theorem is referenced by:  bj-dvelimdv1  33744