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Theorem bj-nfimt 32025
 Description: Closed form of nfim 1813. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nfimt (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑𝜓)))

Proof of Theorem bj-nfimt
StepHypRef Expression
1 19.35 1794 . . 3 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 df-nf 1701 . . . . . . 7 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
32biimpi 205 . . . . . 6 (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
43imim1d 80 . . . . 5 (Ⅎ𝑥𝜑 → ((∀𝑥𝜑 → ∃𝑥𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓)))
5 df-nf 1701 . . . . . . 7 (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
65biimpi 205 . . . . . 6 (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → ∀𝑥𝜓))
76imim2d 55 . . . . 5 (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∃𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
84, 7syl9 75 . . . 4 (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → ((∀𝑥𝜑 → ∃𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))))
9 19.38 1757 . . . 4 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
108, 9syl8 74 . . 3 (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → ((∀𝑥𝜑 → ∃𝑥𝜓) → ∀𝑥(𝜑𝜓))))
111, 10syl7bi 244 . 2 (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → (∃𝑥(𝜑𝜓) → ∀𝑥(𝜑𝜓))))
12 df-nf 1701 . 2 (Ⅎ𝑥(𝜑𝜓) ↔ (∃𝑥(𝜑𝜓) → ∀𝑥(𝜑𝜓)))
1311, 12syl6ibr 241 1 (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728 This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701 This theorem is referenced by:  bj-nfimt2  32026  bj-dvelimdv1  32028
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