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Mirrors > Home > ILE Home > Th. List > sbco2v | GIF version |
Description: Version of sbco2 1975 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 1957 | . 2 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
3 | sbequ 1850 | . 2 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
4 | 2, 3 | sbiev 1802 | 1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 Ⅎwnf 1470 [wsb 1772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 |
This theorem depends on definitions: df-bi 117 df-nf 1471 df-sb 1773 |
This theorem is referenced by: (None) |
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