Theorem wl-euae 36979

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 2559	. 2 ⊢ (∃!𝑥⊤ ↔ (∃𝑥⊤ ∧ ∃*𝑥⊤))
2		extru 1972	. . 3 ⊢ ∃𝑥⊤
3	2	biantrur 530	. 2 ⊢ (∃*𝑥⊤ ↔ (∃𝑥⊤ ∧ ∃*𝑥⊤))
4		wl-moae 36978	. 2 ⊢ (∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
5	1, 3, 4	3bitr2i 299	1 ⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)

Syntax hints:   ↔ wb 205   ∧ wa 395  ∀wal 1532  ⊤wtru 1535  ∃wex 1774  ∃*wmo 2528  ∃!weu 2558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-mo 2530  df-eu 2559

This theorem is referenced by: (None)