![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 1866 | . . 3 ⊢ Ⅎ𝑧[𝑤 / 𝑥]𝜑 |
3 | 2 | nfsbxy 1866 | . 2 ⊢ Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 |
4 | ax-17 1464 | . . . 4 ⊢ (𝜑 → ∀𝑤𝜑) | |
5 | 4 | sbco2v 1869 | . . 3 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
6 | 5 | nfbii 1407 | . 2 ⊢ (Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑) |
7 | 3, 6 | mpbi 143 | 1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
Colors of variables: wff set class |
Syntax hints: Ⅎwnf 1394 [wsb 1692 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 |
This theorem depends on definitions: df-bi 115 df-nf 1395 df-sb 1693 |
This theorem is referenced by: hbsb 1871 sbco2yz 1885 sbcomxyyz 1894 hbsbd 1906 nfsb4or 1947 sb8eu 1961 nfeu 1967 cbvab 2210 cbvralf 2584 cbvrexf 2585 cbvreu 2588 cbvralsv 2601 cbvrexsv 2602 cbvrab 2617 cbvreucsf 2992 cbvrabcsf 2993 cbvopab1 3911 cbvmptf 3932 cbvmpt 3933 ralxpf 4582 rexxpf 4583 cbviota 4985 sb8iota 4987 cbvriota 5618 dfoprab4f 5963 |
Copyright terms: Public domain | W3C validator |