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Mirrors > Home > ILE Home > Th. List > sbco3v | GIF version |
Description: Version of sbco3 1954 with a distinct variable constraint between 𝑥 and 𝑦. (Contributed by Jim Kingdon, 19-Feb-2018.) |
Ref | Expression |
---|---|
sbco3v | ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfs1v 1919 | . . . 4 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
2 | 1 | nfri 1499 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) |
3 | 2 | sbco2vh 1925 | . 2 ⊢ ([𝑧 / 𝑥][𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑) |
4 | sbco 1948 | . . 3 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑) | |
5 | 4 | sbbii 1745 | . 2 ⊢ ([𝑧 / 𝑥][𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑) |
6 | 3, 5 | bitr3i 185 | 1 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 [wsb 1742 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-sb 1743 |
This theorem is referenced by: sbcomv 1951 |
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