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Theorem spaev 2057
Description: A special instance of sp 2183 applied to an equality with a disjoint variable condition. Unlike the more general sp 2183, we can prove this without ax-12 2178. Instance of aeveq 2061.

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 2032 . 2 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑧 = 𝑦))
21spw 2041 1 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by:  aevlem0  2059
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