Theorem wl-euae 38153

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

Assertion

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

Distinct variable group:   𝑥,𝑦

Proof of Theorem wl-euae

Step	Hyp	Ref	Expression
1		df-eu 2597	. 2 ⊢ (∃!𝑥⊤ ↔ (∃𝑥⊤ ∧ ∃*𝑥⊤))
2		extru 2005	. . 3 ⊢ ∃𝑥⊤
3	2	biantrur 539	. 2 ⊢ (∃*𝑥⊤ ↔ (∃𝑥⊤ ∧ ∃*𝑥⊤))
4		wl-moae 38152	. 2 ⊢ (∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
5	1, 3, 4	3bitr2i 302	1 ⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)

Syntax hints:   ↔ wb 209   ∧ wa 400  ∀wal 1568  ⊤wtru 1571  ∃wex 1809  ∃*wmo 2565  ∃!weu 2596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-mo 2567  df-eu 2597

This theorem is referenced by: (None)