Theorem bj-nfimt 37130

Description: Closed form of nfim 1923 and curried (exported) form of nfimt 1922. (Contributed by BJ, 20-Oct-2021.) Proof should not use 19.35 1904. (Proof modification is discouraged.)

Assertion

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

Proof of Theorem bj-nfimt

Step	Hyp	Ref	Expression
1		id 23	. . . . 5 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
2	1	nfrd 1818	. . . 4 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
3		bj-eximcom 37124	. . . 4 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
4	2, 3	syl9 78	. . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓)))
5		id 23	. . . . . 6 ⊢ (Ⅎ𝑥𝜓 → Ⅎ𝑥𝜓)
6	5	nfrd 1818	. . . . 5 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → ∀𝑥𝜓))
7	6	imim2d 58	. . . 4 ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∃𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
8		19.38 1866	. . . 4 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
9	7, 8	syl6 36	. . 3 ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∃𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)))
10	4, 9	syl9 78	. 2 ⊢ (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → (∃𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))))
11		df-nf 1811	. 2 ⊢ (Ⅎ𝑥(𝜑 → 𝜓) ↔ (∃𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)))
12	10, 11	imbitrrdi 255	1 ⊢ (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑 → 𝜓)))

Syntax hints:   → wi 4  ∀wal 1565  ∃wex 1806  Ⅎwnf 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836

This theorem depends on definitions:  df-bi 210  df-ex 1807  df-nf 1811

This theorem is referenced by:  bj-dvelimdv1  37372