![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > sb7af | GIF version |
Description: An alternate definition of proper substitution df-sb 1763. Similar to dfsb7a 1994 but does not require that 𝜑 and 𝑧 be distinct. Similar to sb7f 1992 in that it involves a dummy variable 𝑧, but expressed in terms of ∀ rather than ∃. (Contributed by Jim Kingdon, 5-Feb-2018.) |
Ref | Expression |
---|---|
sb7af.1 | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
sb7af | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb6 1886 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑)) | |
2 | 1 | sbbii 1765 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧]∀𝑥(𝑥 = 𝑧 → 𝜑)) |
3 | sb7af.1 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
4 | 3 | sbco2 1965 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
5 | sb6 1886 | . 2 ⊢ ([𝑦 / 𝑧]∀𝑥(𝑥 = 𝑧 → 𝜑) ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | |
6 | 2, 4, 5 | 3bitr3i 210 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1351 Ⅎwnf 1460 [wsb 1762 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 |
This theorem is referenced by: dfsb7a 1994 |
Copyright terms: Public domain | W3C validator |