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

Theorem dfsb7 2281
Description: An alternate definition of proper substitution df-sb 2066. By introducing a dummy variable 𝑦 in the definiens, we are able to eliminate any distinct variable restrictions among the variables 𝑡, 𝑥, and 𝜑 of the definiendum. No distinct variable conflicts arise because 𝑦 effectively insulates 𝑡 from 𝑥. To achieve this, we use a chain of two substitutions in the form of sb5 2272, first 𝑦 for 𝑥 then 𝑡 for 𝑦. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2800. Theorem sb7h 2565 provides a version where 𝜑 and 𝑦 don't have to be distinct. (Contributed by NM, 28-Jan-2004.) Revise df-sb 2066. (Revised by BJ, 25-Dec-2020.) (Proof shortened by Wolf Lammen, 3-Sep-2023.)
Assertion
Ref Expression
dfsb7 ([𝑡 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
Distinct variable groups:   𝑦,𝑡   𝑥,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem dfsb7
StepHypRef Expression
1 sb56 2273 . 2 (∃𝑦(𝑦 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 sb56 2273 . . . 4 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
32anbi2i 624 . . 3 ((𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ (𝑦 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑦𝜑)))
43exbii 1844 . 2 (∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑦𝜑)))
5 df-sb 2066 . 2 ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
61, 4, 53bitr4ri 306 1 ([𝑡 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1531  wex 1776  [wsb 2065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-12 2172
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-sb 2066
This theorem is referenced by:  sbn  2283
  Copyright terms: Public domain W3C validator