Theorem nfim1 2237

Description: A closed form of nfim 1929. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1817 changed. (Revised by Wolf Lammen, 18-Sep-2021.)

Hypotheses

Ref	Expression
nfim1.1	⊢ Ⅎ𝑥𝜑
nfim1.2	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfim1	⊢ Ⅎ𝑥(𝜑 → 𝜓)

Proof of Theorem nfim1

Step	Hyp	Ref	Expression
1		nfim1.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		nf3 1819	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
3	1, 2	mpbi 233	. 2 ⊢ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)
4		nftht 1825	. . . 4 ⊢ (∀𝑥𝜑 → Ⅎ𝑥𝜑)
5		nfim1.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
6	5	sps 2223	. . . 4 ⊢ (∀𝑥𝜑 → Ⅎ𝑥𝜓)
7	4, 6	nfimd 1927	. . 3 ⊢ (∀𝑥𝜑 → Ⅎ𝑥(𝜑 → 𝜓))
8		pm2.21 124	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
9	8	alimi 1844	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝜓))
10		nftht 1825	. . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → Ⅎ𝑥(𝜑 → 𝜓))
11	9, 10	syl 18	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → Ⅎ𝑥(𝜑 → 𝜓))
12	7, 11	jaoi 871	. 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → Ⅎ𝑥(𝜑 → 𝜓))
13	3, 12	ax-mp 5	1 ⊢ Ⅎ𝑥(𝜑 → 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∨ wo 861  ∀wal 1568  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2215

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by:  nfan1  2238  sbiedw  2348  cbv1v  2367  cbv1  2433  dvelimdf  2480  sbied  2534  sbco2d  2543  ichnfimlem  48348