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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nfimt | Structured version Visualization version GIF version |
Description: Closed form of nfim 1903 and curried (exported) form of nfimt 1902. (Contributed by BJ, 20-Oct-2021.) |
Ref | Expression |
---|---|
bj-nfimt | ⊢ (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.35 1884 | . . . 4 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
2 | id 22 | . . . . . 6 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑) | |
3 | 2 | nfrd 1798 | . . . . 5 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑)) |
4 | 3 | imim1d 82 | . . . 4 ⊢ (Ⅎ𝑥𝜑 → ((∀𝑥𝜑 → ∃𝑥𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))) |
5 | 1, 4 | syl5bi 245 | . . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))) |
6 | id 22 | . . . . . 6 ⊢ (Ⅎ𝑥𝜓 → Ⅎ𝑥𝜓) | |
7 | 6 | nfrd 1798 | . . . . 5 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → ∀𝑥𝜓)) |
8 | 7 | imim2d 57 | . . . 4 ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∃𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))) |
9 | 19.38 1845 | . . . 4 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)) | |
10 | 8, 9 | syl6 35 | . . 3 ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∃𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))) |
11 | 5, 10 | syl9 77 | . 2 ⊢ (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → (∃𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)))) |
12 | df-nf 1791 | . 2 ⊢ (Ⅎ𝑥(𝜑 → 𝜓) ↔ (∃𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))) | |
13 | 11, 12 | syl6ibr 255 | 1 ⊢ (Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 ∃wex 1786 Ⅎwnf 1790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 |
This theorem depends on definitions: df-bi 210 df-ex 1787 df-nf 1791 |
This theorem is referenced by: bj-dvelimdv1 34692 |
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