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Mirrors > Home > ILE Home > Th. List > nfimd | GIF version |
Description: If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓 → 𝜒). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) |
Ref | Expression |
---|---|
nfimd.1 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
nfimd.2 | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
Ref | Expression |
---|---|
nfimd | ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfimd.1 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
2 | nfimd.2 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
3 | nfnf1 1524 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝜓 | |
4 | 3 | nfri 1500 | . . . 4 ⊢ (Ⅎ𝑥𝜓 → ∀𝑥Ⅎ𝑥𝜓) |
5 | nfnf1 1524 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝜒 | |
6 | 5 | nfri 1500 | . . . 4 ⊢ (Ⅎ𝑥𝜒 → ∀𝑥Ⅎ𝑥𝜒) |
7 | nfr 1499 | . . . . . 6 ⊢ (Ⅎ𝑥𝜒 → (𝜒 → ∀𝑥𝜒)) | |
8 | 7 | imim2d 54 | . . . . 5 ⊢ (Ⅎ𝑥𝜒 → ((𝜓 → 𝜒) → (𝜓 → ∀𝑥𝜒))) |
9 | 19.21t 1562 | . . . . . 6 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜓 → 𝜒) ↔ (𝜓 → ∀𝑥𝜒))) | |
10 | 9 | biimprd 157 | . . . . 5 ⊢ (Ⅎ𝑥𝜓 → ((𝜓 → ∀𝑥𝜒) → ∀𝑥(𝜓 → 𝜒))) |
11 | 8, 10 | syl9r 73 | . . . 4 ⊢ (Ⅎ𝑥𝜓 → (Ⅎ𝑥𝜒 → ((𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒)))) |
12 | 4, 6, 11 | alrimdh 1456 | . . 3 ⊢ (Ⅎ𝑥𝜓 → (Ⅎ𝑥𝜒 → ∀𝑥((𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒)))) |
13 | df-nf 1438 | . . 3 ⊢ (Ⅎ𝑥(𝜓 → 𝜒) ↔ ∀𝑥((𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒))) | |
14 | 12, 13 | syl6ibr 161 | . 2 ⊢ (Ⅎ𝑥𝜓 → (Ⅎ𝑥𝜒 → Ⅎ𝑥(𝜓 → 𝜒))) |
15 | 1, 2, 14 | sylc 62 | 1 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1330 Ⅎwnf 1437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-gen 1426 ax-4 1488 ax-ial 1515 ax-i5r 1516 |
This theorem depends on definitions: df-bi 116 df-nf 1438 |
This theorem is referenced by: nfbid 1568 dvelimALT 1986 dvelimfv 1987 dvelimor 1994 nfmod 2017 nfraldxy 2470 nfixpxy 6619 cbvrald 13166 |
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