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Mirrors > Home > ILE Home > Th. List > sbco2vh | GIF version |
Description: This is a version of sbco2 1938 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 1917 | . . 3 ⊢ ([𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑) |
3 | 2 | sbbii 1738 | . 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑤 / 𝑥]𝜑) |
4 | ax-17 1506 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 → ∀𝑤[𝑧 / 𝑥]𝜑) | |
5 | 4 | sbco2vlem 1917 | . 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑) |
6 | ax-17 1506 | . . 3 ⊢ (𝜑 → ∀𝑤𝜑) | |
7 | 6 | sbco2vlem 1917 | . 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 1329 [wsb 1735 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 |
This theorem is referenced by: nfsb 1919 equsb3 1924 sbn 1925 sbim 1926 sbor 1927 sban 1928 sbco2vd 1940 sbco3v 1942 sbcom2v2 1961 sbcom2 1962 dfsb7 1966 sb7f 1967 sbal 1975 sbal1 1977 sbex 1979 |
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