Theorem wl-euae

Description: Two ways to express "exactly one thing exists" . (Contributed by Wolf Lammen, 5-Mar-2023)

	Ref	Expression
Assertion	wl-euae	⊢ ( ∃! 𝑥 ⊤ ↔ ∀ 𝑥 𝑥 = 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		df-eu	⊢ ( ∃! 𝑥 ⊤ ↔ ( ∃ 𝑥 ⊤ ∧ ∃* 𝑥 ⊤ ) )
2		extru	⊢ ∃ 𝑥 ⊤
3	2	biantrur	⊢ ( ∃* 𝑥 ⊤ ↔ ( ∃ 𝑥 ⊤ ∧ ∃* 𝑥 ⊤ ) )
4		wl-moae	⊢ ( ∃* 𝑥 ⊤ ↔ ∀ 𝑥 𝑥 = 𝑦 )
5	1 3 4	3bitr2i	⊢ ( ∃! 𝑥 ⊤ ↔ ∀ 𝑥 𝑥 = 𝑦 )