Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nfsb4or | GIF version |
Description: A variable not free remains so after substitution with a distinct variable. (Contributed by Jim Kingdon, 11-May-2018.) |
Ref | Expression |
---|---|
nfsb4or.1 | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
nfsb4or | ⊢ (∀𝑧 𝑧 = 𝑦 ∨ Ⅎ𝑧[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfsb4or.1 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | nfsb 1863 | . 2 ⊢ Ⅎ𝑧[𝑤 / 𝑥]𝜑 |
3 | sbequ 1761 | . 2 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
4 | 2, 3 | dvelimor 1935 | 1 ⊢ (∀𝑧 𝑧 = 𝑦 ∨ Ⅎ𝑧[𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ∨ wo 661 ∀wal 1282 Ⅎwnf 1389 [wsb 1685 |
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 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |