Theorem nfim1 2191

Description: A closed form of nfim 1898. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1785 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 1787	. . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
3	1, 2	mpbi 229	. 2 ⊢ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)
4		nftht 1793	. . . 4 ⊢ (∀𝑥𝜑 → Ⅎ𝑥𝜑)
5		nfim1.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
6	5	sps 2177	. . . 4 ⊢ (∀𝑥𝜑 → Ⅎ𝑥𝜓)
7	4, 6	nfimd 1896	. . 3 ⊢ (∀𝑥𝜑 → Ⅎ𝑥(𝜑 → 𝜓))
8		pm2.21 123	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
9	8	alimi 1812	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝜓))
10		nftht 1793	. . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → Ⅎ𝑥(𝜑 → 𝜓))
11	9, 10	syl 17	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → Ⅎ𝑥(𝜑 → 𝜓))
12	7, 11	jaoi 854	. 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → Ⅎ𝑥(𝜑 → 𝜓))
13	3, 12	ax-mp 5	1 ⊢ Ⅎ𝑥(𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 844  ∀wal 1538  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-12 2170

This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1781  df-nf 1785

This theorem is referenced by:  nfan1  2192  sbiedw  2308  cbv1v  2331  cbv1  2400  dvelimdf  2447  sbied  2501  sbco2d  2510  nfabdwOLD  2926  ichnfimlem  46442