Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > sbco2yz | GIF version |
Description: This is a version of sbco2 1839 where 𝑧 is distinct from 𝑦. It is a lemma on the way to proving sbco2 1839 which has no distinct variable constraints. (Contributed by Jim Kingdon, 19-Mar-2018.) |
Ref | Expression |
---|---|
sbco2yz.1 | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
sbco2yz | ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbco2yz.1 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | nfsb 1822 | . . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
3 | 2 | nfri 1412 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑) |
4 | sbequ 1721 | . 2 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
5 | 3, 4 | sbieh 1673 | 1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 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: sbco2h 1838 |
Copyright terms: Public domain | W3C validator |