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| Mirrors > Home > ILE Home > Th. List > sbco2vh | GIF version | ||
| Description: This is a version of sbco2 1992 where 𝑧 is distinct from 𝑥. (Contributed by Jim Kingdon, 12-Feb-2018.) |
| Ref | Expression |
|---|---|
| sbco2vh.1 | ⊢ (𝜑 → ∀𝑧𝜑) |
| Ref | Expression |
|---|---|
| sbco2vh | ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbco2vh.1 | . . . 4 ⊢ (𝜑 → ∀𝑧𝜑) | |
| 2 | 1 | sbco2vlem 1971 | . . 3 ⊢ ([𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑) |
| 3 | 2 | sbbii 1787 | . 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑤 / 𝑥]𝜑) |
| 4 | ax-17 1548 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 → ∀𝑤[𝑧 / 𝑥]𝜑) | |
| 5 | 4 | sbco2vlem 1971 | . 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑) |
| 6 | ax-17 1548 | . . 3 ⊢ (𝜑 → ∀𝑤𝜑) | |
| 7 | 6 | sbco2vlem 1971 | . 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
| 8 | 3, 5, 7 | 3bitr3i 210 | 1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ↔ wb 105 ∀wal 1370 [wsb 1784 |
| 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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 |
| This theorem depends on definitions: df-bi 117 df-nf 1483 df-sb 1785 |
| This theorem is referenced by: nfsb 1973 equsb3 1978 sbn 1979 sbim 1980 sbor 1981 sban 1982 sbco2vd 1994 sbco3v 1996 sbcom2v2 2013 sbcom2 2014 dfsb7 2018 sb7f 2019 sbal 2027 sbal1 2029 sbex 2031 |
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