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Mirrors > Home > ILE Home > Th. List > dfsb7a | GIF version |
Description: An alternate definition of proper substitution df-sb 1736. Similar to dfsb7 1964 in that it involves a dummy variable 𝑧, but expressed in terms of ∀ rather than ∃. For a version which only requires Ⅎ𝑧𝜑 rather than 𝑧 and 𝜑 being distinct, see sb7af 1966. (Contributed by Jim Kingdon, 5-Feb-2018.) |
Ref | Expression |
---|---|
dfsb7a | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1508 | . 2 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | sb7af 1966 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 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-bndl 1486 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: (None) |
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