MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfsb7 Structured version   Visualization version   GIF version

Theorem dfsb7 2575
Description: An alternate definition of proper substitution df-sb 2065. 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 2300, 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 2786. Theorem sb7h 2574 provides a version where 𝜑 and 𝑧 don't have to be distinct. (Contributed by NM, 28-Jan-2004.)
Assertion
Ref Expression
dfsb7 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))
Distinct variable groups:   𝑦,𝑧   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem dfsb7
StepHypRef Expression
1 nfv 2010 . 2 𝑧𝜑
21sb7f 2573 1 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385  wex 1875  [wsb 2064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator