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Theorem nfimt 1819
Description: Closed form of nfim 1823 and curried (exported) form of nfimt2 1820. (Contributed by BJ, 20-Oct-2021.)
Assertion
Ref Expression
nfimt (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑𝜓)))

Proof of Theorem nfimt
StepHypRef Expression
1 19.35 1803 . . 3 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 df-nf 1708 . . . . . . 7 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
32biimpi 206 . . . . . 6 (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
43imim1d 82 . . . . 5 (Ⅎ𝑥𝜑 → ((∀𝑥𝜑 → ∃𝑥𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓)))
5 df-nf 1708 . . . . . . 7 (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
65biimpi 206 . . . . . 6 (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → ∀𝑥𝜓))
76imim2d 57 . . . . 5 (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∃𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
84, 7syl9 77 . . . 4 (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → ((∀𝑥𝜑 → ∃𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))))
9 19.38 1764 . . . 4 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
108, 9syl8 76 . . 3 (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → ((∀𝑥𝜑 → ∃𝑥𝜓) → ∀𝑥(𝜑𝜓))))
111, 10syl7bi 245 . 2 (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → (∃𝑥(𝜑𝜓) → ∀𝑥(𝜑𝜓))))
12 df-nf 1708 . 2 (Ⅎ𝑥(𝜑𝜓) ↔ (∃𝑥(𝜑𝜓) → ∀𝑥(𝜑𝜓)))
1311, 12syl6ibr 242 1 (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1479  wex 1702  wnf 1706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735
This theorem depends on definitions:  df-bi 197  df-ex 1703  df-nf 1708
This theorem is referenced by:  nfimt2  1820  bj-dvelimdv1  32810
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