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Mirrors > Home > ILE Home > Th. List > nfexd | GIF version |
Description: If 𝑥 is not free in 𝜑, it is not free in ∃𝑦𝜑. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 7-Feb-2018.) |
Ref | Expression |
---|---|
nfald.1 | ⊢ Ⅎ𝑦𝜑 |
nfald.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfexd | ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfald.1 | . . . . . . 7 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfri 1458 | . . . . . 6 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | nfald.2 | . . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
4 | df-nf 1396 | . . . . . . 7 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓)) | |
5 | 3, 4 | sylib 121 | . . . . . 6 ⊢ (𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓)) |
6 | 2, 5 | alrimih 1404 | . . . . 5 ⊢ (𝜑 → ∀𝑦∀𝑥(𝜓 → ∀𝑥𝜓)) |
7 | alcom 1413 | . . . . 5 ⊢ (∀𝑦∀𝑥(𝜓 → ∀𝑥𝜓) ↔ ∀𝑥∀𝑦(𝜓 → ∀𝑥𝜓)) | |
8 | 6, 7 | sylib 121 | . . . 4 ⊢ (𝜑 → ∀𝑥∀𝑦(𝜓 → ∀𝑥𝜓)) |
9 | exim 1536 | . . . . 5 ⊢ (∀𝑦(𝜓 → ∀𝑥𝜓) → (∃𝑦𝜓 → ∃𝑦∀𝑥𝜓)) | |
10 | 9 | alimi 1390 | . . . 4 ⊢ (∀𝑥∀𝑦(𝜓 → ∀𝑥𝜓) → ∀𝑥(∃𝑦𝜓 → ∃𝑦∀𝑥𝜓)) |
11 | 8, 10 | syl 14 | . . 3 ⊢ (𝜑 → ∀𝑥(∃𝑦𝜓 → ∃𝑦∀𝑥𝜓)) |
12 | 19.12 1601 | . . . . 5 ⊢ (∃𝑦∀𝑥𝜓 → ∀𝑥∃𝑦𝜓) | |
13 | 12 | imim2i 12 | . . . 4 ⊢ ((∃𝑦𝜓 → ∃𝑦∀𝑥𝜓) → (∃𝑦𝜓 → ∀𝑥∃𝑦𝜓)) |
14 | 13 | alimi 1390 | . . 3 ⊢ (∀𝑥(∃𝑦𝜓 → ∃𝑦∀𝑥𝜓) → ∀𝑥(∃𝑦𝜓 → ∀𝑥∃𝑦𝜓)) |
15 | 11, 14 | syl 14 | . 2 ⊢ (𝜑 → ∀𝑥(∃𝑦𝜓 → ∀𝑥∃𝑦𝜓)) |
16 | df-nf 1396 | . 2 ⊢ (Ⅎ𝑥∃𝑦𝜓 ↔ ∀𝑥(∃𝑦𝜓 → ∀𝑥∃𝑦𝜓)) | |
17 | 15, 16 | sylibr 133 | 1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1288 Ⅎwnf 1395 ∃wex 1427 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-4 1446 ax-ial 1473 |
This theorem depends on definitions: df-bi 116 df-nf 1396 |
This theorem is referenced by: nfsbxy 1867 nfsbxyt 1868 nfeudv 1964 nfmod 1966 nfeld 2245 nfrexdxy 2412 |
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