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Mirrors > Home > ILE Home > Th. List > sbco2v | GIF version |
Description: This is a version of sbco2 1839 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 1820 | . . 3 ⊢ ([𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑) |
3 | 2 | sbbii 1648 | . 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑤 / 𝑥]𝜑) |
4 | ax-17 1419 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 → ∀𝑤[𝑧 / 𝑥]𝜑) | |
5 | 4 | sbco2vlem 1820 | . 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑) |
6 | ax-17 1419 | . . 3 ⊢ (𝜑 → ∀𝑤𝜑) | |
7 | 6 | sbco2vlem 1820 | . 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
8 | 3, 5, 7 | 3bitr3i 199 | 1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1241 [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-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: nfsb 1822 equsb3 1825 sbn 1826 sbim 1827 sbor 1828 sban 1829 sbco2vd 1841 sbco3v 1843 sbcom2v2 1862 sbcom2 1863 dfsb7 1867 sb7f 1868 sbal 1876 sbal1 1878 sbex 1880 |
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