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Mirrors > Home > MPE Home > Th. List > dfsb7 | Structured version Visualization version GIF version |
Description: An alternate definition of proper substitution df-sb 1881. By introducing a dummy variable 𝑧 in the definiens, we are able to eliminate any distinct variable restrictions among the variables 𝑥, 𝑦, and 𝜑 of the definiendum. No distinct variable conflicts arise because 𝑧 effectively insulates 𝑥 from 𝑦. To achieve this, we use a chain of two substitutions in the form of sb5 2430, first 𝑧 for 𝑥 then 𝑦 for 𝑧. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2609. Theorem sb7h 2454 provides a version where 𝜑 and 𝑧 don't have to be distinct. (Contributed by NM, 28-Jan-2004.) |
Ref | Expression |
---|---|
dfsb7 | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1843 | . 2 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | sb7f 2453 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∃wex 1704 [wsb 1880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 |
This theorem is referenced by: (None) |
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