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Mirrors > Home > ILE Home > Th. List > sbco2v | GIF version |
Description: Version of sbco2 1936 with disjoint variable conditions. (Contributed by Wolf Lammen, 29-Apr-2023.) |
Ref | Expression |
---|---|
sbco2v.1 | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
sbco2v | ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbco2v.1 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | nfsbv 1918 | . 2 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
3 | sbequ 1812 | . 2 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
4 | 2, 3 | sbiev 1765 | 1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 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: cbvabw 2260 |
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