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Mirrors > Home > ILE Home > Th. List > nfald | GIF version |
Description: If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.) |
Ref | Expression |
---|---|
nfald.1 | ⊢ Ⅎ𝑦𝜑 |
nfald.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfald | ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfald.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfri 1412 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | nfald.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
4 | 2, 3 | alrimih 1358 | . 2 ⊢ (𝜑 → ∀𝑦Ⅎ𝑥𝜓) |
5 | nfnf1 1436 | . . . 4 ⊢ Ⅎ𝑥Ⅎ𝑥𝜓 | |
6 | 5 | nfal 1468 | . . 3 ⊢ Ⅎ𝑥∀𝑦Ⅎ𝑥𝜓 |
7 | hba1 1433 | . . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜓 → ∀𝑦∀𝑦Ⅎ𝑥𝜓) | |
8 | sp 1401 | . . . . 5 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑥𝜓) | |
9 | 8 | nfrd 1413 | . . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓)) |
10 | 7, 9 | hbald 1380 | . . 3 ⊢ (∀𝑦Ⅎ𝑥𝜓 → (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓)) |
11 | 6, 10 | nfd 1416 | . 2 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑥∀𝑦𝜓) |
12 | 4, 11 | syl 14 | 1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀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-7 1337 ax-gen 1338 ax-4 1400 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-nf 1350 |
This theorem is referenced by: dvelimALT 1886 dvelimfv 1887 nfeudv 1915 nfeqd 2192 nfraldxy 2356 nfiotadxy 4870 bdsepnft 10007 strcollnft 10109 |
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