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Theorem nfim1 2192
Description: A closed form of nfim 1899. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1787 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 1789 . . 3 (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
31, 2mpbi 229 . 2 (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)
4 nftht 1795 . . . 4 (∀𝑥𝜑 → Ⅎ𝑥𝜑)
5 nfim1.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
65sps 2178 . . . 4 (∀𝑥𝜑 → Ⅎ𝑥𝜓)
74, 6nfimd 1897 . . 3 (∀𝑥𝜑 → Ⅎ𝑥(𝜑𝜓))
8 pm2.21 123 . . . . 5 𝜑 → (𝜑𝜓))
98alimi 1814 . . . 4 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝜓))
10 nftht 1795 . . . 4 (∀𝑥(𝜑𝜓) → Ⅎ𝑥(𝜑𝜓))
119, 10syl 17 . . 3 (∀𝑥 ¬ 𝜑 → Ⅎ𝑥(𝜑𝜓))
127, 11jaoi 854 . 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 844  wal 1537  wnf 1786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1783  df-nf 1787
This theorem is referenced by:  nfan1  2193  sbiedw  2310  cbv1v  2333  cbv1  2402  dvelimdf  2449  sbied  2507  sbco2d  2516  nfabdwOLD  2931  ichnfimlem  44915
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