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Mirrors > Home > ILE Home > Th. List > sbco2vh | GIF version |
Description: This is a version of sbco2 1939 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 1918 | . . 3 ⊢ ([𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑) |
3 | 2 | sbbii 1739 | . 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑤 / 𝑥]𝜑) |
4 | ax-17 1507 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 → ∀𝑤[𝑧 / 𝑥]𝜑) | |
5 | 4 | sbco2vlem 1918 | . 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑) |
6 | ax-17 1507 | . . 3 ⊢ (𝜑 → ∀𝑤𝜑) | |
7 | 6 | sbco2vlem 1918 | . 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 1330 [wsb 1736 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 |
This theorem is referenced by: nfsb 1920 equsb3 1925 sbn 1926 sbim 1927 sbor 1928 sban 1929 sbco2vd 1941 sbco3v 1943 sbcom2v2 1962 sbcom2 1963 dfsb7 1967 sb7f 1968 sbal 1976 sbal1 1978 sbex 1980 |
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