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 1917 | . 2 ⊢ Ⅎ𝑧[𝑤 / 𝑥]𝜑 |
3 | sbequ 1812 | . 2 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
4 | 2, 3 | dvelimor 1991 | 1 ⊢ (∀𝑧 𝑧 = 𝑦 ∨ Ⅎ𝑧[𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ∨ wo 697 ∀wal 1329 Ⅎwnf 1436 [wsb 1735 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |