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Mirrors > Home > ILE Home > Th. List > sb7af | GIF version |
Description: An alternate definition of proper substitution df-sb 1700. Similar to dfsb7a 1925 but does not require that 𝜑 and 𝑧 be distinct. Similar to sb7f 1923 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 1821 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑)) | |
2 | 1 | sbbii 1702 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧]∀𝑥(𝑥 = 𝑧 → 𝜑)) |
3 | sb7af.1 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
4 | 3 | sbco2 1894 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
5 | sb6 1821 | . 2 ⊢ ([𝑦 / 𝑧]∀𝑥(𝑥 = 𝑧 → 𝜑) ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | |
6 | 2, 4, 5 | 3bitr3i 209 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1294 Ⅎwnf 1401 [wsb 1699 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 |
This theorem depends on definitions: df-bi 116 df-nf 1402 df-sb 1700 |
This theorem is referenced by: dfsb7a 1925 |
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