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

Theorem spaev 2095
Description: A special instance of sp 2167 applied to an equality with a disjoint variable condition. Unlike the more general sp 2167, we can prove this without ax-12 2163. Instance of aeveq 2099.

The antecedent 𝑥𝑥 = 𝑦 with distinct 𝑥 and 𝑦 is a characteristic of a degenerate universe, in which just one object exists. Actually more than one object may still exist, but if so, we give up on equality as a discriminating term.

Separating this degenerate case from a richer universe, where inequality is possible, is a common proof idea. The name of this theorem follows a convention, where the condition 𝑥𝑥 = 𝑦 is denoted by 'aev', a shorthand for 'all equal, with a distinct variable condition'. (Contributed by Wolf Lammen, 14-Mar-2021.)

Ref Expression
spaev (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
Distinct variable group:   𝑥,𝑦

Proof of Theorem spaev
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 equequ1 2072 . 2 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑧 = 𝑦))
21spw 2084 1 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055
This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824
This theorem is referenced by:  aevlem0  2097
  Copyright terms: Public domain W3C validator