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Theorem bj-nfimt 33053
Description: Closed form of nfim 1995 and curried (exported) form of nfimt 1993. (Contributed by BJ, 20-Oct-2021.)
Assertion
Ref Expression
bj-nfimt (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑𝜓)))

Proof of Theorem bj-nfimt
StepHypRef Expression
1 19.35 1976 . . . 4 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 id 22 . . . . . 6 (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
32nfrd 1886 . . . . 5 (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
43imim1d 82 . . . 4 (Ⅎ𝑥𝜑 → ((∀𝑥𝜑 → ∃𝑥𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓)))
51, 4syl5bi 233 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓)))
6 id 22 . . . . . 6 (Ⅎ𝑥𝜓 → Ⅎ𝑥𝜓)
76nfrd 1886 . . . . 5 (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → ∀𝑥𝜓))
87imim2d 57 . . . 4 (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∃𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
9 19.38 1933 . . . 4 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
108, 9syl6 35 . . 3 (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∃𝑥𝜓) → ∀𝑥(𝜑𝜓)))
115, 10syl9 77 . 2 (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → (∃𝑥(𝜑𝜓) → ∀𝑥(𝜑𝜓))))
12 df-nf 1879 . 2 (Ⅎ𝑥(𝜑𝜓) ↔ (∃𝑥(𝜑𝜓) → ∀𝑥(𝜑𝜓)))
1311, 12syl6ibr 243 1 (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1650  wex 1874  wnf 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904
This theorem depends on definitions:  df-bi 198  df-ex 1875  df-nf 1879
This theorem is referenced by:  bj-dvelimdv1  33263
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