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Mirrors > Home > ILE Home > Th. List > sb7af | GIF version |
Description: An alternate definition of proper substitution df-sb 1774. Similar to dfsb7a 2006 but does not require that 𝜑 and 𝑧 be distinct. Similar to sb7f 2004 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 1898 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑)) | |
2 | 1 | sbbii 1776 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧]∀𝑥(𝑥 = 𝑧 → 𝜑)) |
3 | sb7af.1 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
4 | 3 | sbco2 1977 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
5 | sb6 1898 | . 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 1362 Ⅎwnf 1471 [wsb 1773 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 |
This theorem is referenced by: dfsb7a 2006 |
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