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Mirrors > Home > ILE Home > Th. List > nfim1 | GIF version |
Description: A closed form of nfim 1464. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |
Ref | Expression |
---|---|
nfim1.1 | ⊢ Ⅎ𝑥𝜑 |
nfim1.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfim1 | ⊢ Ⅎ𝑥(𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfim1.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | nfri 1412 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) |
3 | nfim1.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
4 | 3 | nfrd 1413 | . . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) |
5 | 2, 4 | hbim1 1462 | . 2 ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)) |
6 | 5 | nfi 1351 | 1 ⊢ Ⅎ𝑥(𝜑 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Ⅎwnf 1349 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-4 1400 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-nf 1350 |
This theorem is referenced by: nfim 1464 cbv1 1632 hbsbd 1858 |
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