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Mirrors > Home > ILE Home > Th. List > nfan1 | GIF version |
Description: A closed form of nfan 1457. (Contributed by Mario Carneiro, 3-Oct-2016.) |
Ref | Expression |
---|---|
nfan1.1 | ⊢ Ⅎ𝑥𝜑 |
nfan1.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfan1 | ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfan1.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
2 | 1 | nfrd 1413 | . . . 4 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) |
3 | 2 | imdistani 419 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ ∀𝑥𝜓)) |
4 | nfan1.1 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
5 | 4 | nfri 1412 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) |
6 | 5 | 19.28h 1454 | . . 3 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) |
7 | 3, 6 | sylibr 137 | . 2 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓)) |
8 | 7 | nfi 1351 | 1 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1241 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 |
This theorem depends on definitions: df-bi 110 df-nf 1350 |
This theorem is referenced by: nfan 1457 sbcralt 2834 sbcrext 2835 csbiebt 2886 riota5f 5492 |
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