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Theorem nfim1 2221
Description: A closed form of nfim 1977. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1858 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
Hypotheses
Ref Expression
nfim1.1 𝑥𝜑
nfim1.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfim1 𝑥(𝜑𝜓)

Proof of Theorem nfim1
StepHypRef Expression
1 nfim1.1 . . 3 𝑥𝜑
2 nf3 1860 . . 3 (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
31, 2mpbi 220 . 2 (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)
4 nftht 1866 . . . 4 (∀𝑥𝜑 → Ⅎ𝑥𝜑)
5 nfim1.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
65sps 2209 . . . 4 (∀𝑥𝜑 → Ⅎ𝑥𝜓)
74, 6nfimd 1973 . . 3 (∀𝑥𝜑 → Ⅎ𝑥(𝜑𝜓))
8 pm2.21 121 . . . . 5 𝜑 → (𝜑𝜓))
98alimi 1887 . . . 4 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝜓))
10 nftht 1866 . . . 4 (∀𝑥(𝜑𝜓) → Ⅎ𝑥(𝜑𝜓))
119, 10syl 17 . . 3 (∀𝑥 ¬ 𝜑 → Ⅎ𝑥(𝜑𝜓))
127, 11jaoi 846 . 2 ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → Ⅎ𝑥(𝜑𝜓))
133, 12ax-mp 5 1 𝑥(𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 836  wal 1629  wnf 1856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-or 837  df-ex 1853  df-nf 1858
This theorem is referenced by:  nfan1  2222  nfan1OLDOLD  2223  cbv1  2428  dvelimdf  2485  sbied  2556  sbco2d  2563  bj-cbv1v  33058
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