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Mirrors > Home > ILE Home > Th. List > sbco2v | GIF version |
Description: This is a version of sbco2 1894 where 𝑧 is distinct from 𝑥. (Contributed by Jim Kingdon, 12-Feb-2018.) |
Ref | Expression |
---|---|
sbco2v.1 | ⊢ (𝜑 → ∀𝑧𝜑) |
Ref | Expression |
---|---|
sbco2v | ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbco2v.1 | . . . 4 ⊢ (𝜑 → ∀𝑧𝜑) | |
2 | 1 | sbco2vlem 1875 | . . 3 ⊢ ([𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑) |
3 | 2 | sbbii 1702 | . 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑤 / 𝑥]𝜑) |
4 | ax-17 1471 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 → ∀𝑤[𝑧 / 𝑥]𝜑) | |
5 | 4 | sbco2vlem 1875 | . 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑) |
6 | ax-17 1471 | . . 3 ⊢ (𝜑 → ∀𝑤𝜑) | |
7 | 6 | sbco2vlem 1875 | . 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
8 | 3, 5, 7 | 3bitr3i 209 | 1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1294 [wsb 1699 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 |
This theorem depends on definitions: df-bi 116 df-nf 1402 df-sb 1700 |
This theorem is referenced by: nfsb 1877 equsb3 1880 sbn 1881 sbim 1882 sbor 1883 sban 1884 sbco2vd 1896 sbco3v 1898 sbcom2v2 1917 sbcom2 1918 dfsb7 1922 sb7f 1923 sbal 1931 sbal1 1933 sbex 1935 |
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