Theorem spaev 2058

Description: A special instance of sp 2179 applied to an equality with a disjoint variable condition. Unlike the more general sp 2179, we can prove this without ax-12 2174. Instance of aeveq 2062.

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.)

Assertion

Ref	Expression
spaev	⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)

Distinct variable group:   𝑥,𝑦

Proof of Theorem spaev

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equequ1 2031	. 2 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑧 = 𝑦))
2	1	spw 2040	1 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)

Syntax hints:   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1786

This theorem is referenced by:  aevlem0  2060