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Mirrors > Home > ILE Home > Th. List > nfsb | GIF version |
Description: If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.) |
Ref | Expression |
---|---|
nfsb.1 | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
nfsb | ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfsb.1 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | nfsbxy 1818 | . . 3 ⊢ Ⅎ𝑧[𝑤 / 𝑥]𝜑 |
3 | 2 | nfsbxy 1818 | . 2 ⊢ Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 |
4 | ax-17 1419 | . . . 4 ⊢ (𝜑 → ∀𝑤𝜑) | |
5 | 4 | sbco2v 1821 | . . 3 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
6 | 5 | nfbii 1362 | . 2 ⊢ (Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑) |
7 | 3, 6 | mpbi 133 | 1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
Colors of variables: wff set class |
Syntax hints: Ⅎwnf 1349 [wsb 1645 |
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-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 |
This theorem is referenced by: hbsb 1823 sbco2yz 1837 sbcomxyyz 1846 hbsbd 1858 nfsb4or 1899 sb8eu 1913 nfeu 1919 cbvab 2160 cbvralf 2527 cbvrexf 2528 cbvreu 2531 cbvralsv 2544 cbvrexsv 2545 cbvrab 2555 cbvreucsf 2910 cbvrabcsf 2911 cbvopab1 3830 cbvmpt 3851 ralxpf 4482 rexxpf 4483 cbviota 4872 sb8iota 4874 cbvriota 5478 dfoprab4f 5819 |
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