Theorem spaev 1978

Description: A special instance of sp 2053 applied to an equality with a dv condition. Unlike the more general sp 2053, we can prove this without ax-12 2047. Instance of aeveq 1982.

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 1952	. 2 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑧 = 𝑦))
2	1	spw 1967	1 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)

Syntax hints:   → wi 4  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  aevlem0  1980  axc11nlemOLD2  1988