Theorem wl-euae 37471

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 2572	. 2 ⊢ (∃!𝑥⊤ ↔ (∃𝑥⊤ ∧ ∃*𝑥⊤))
2		extru 1975	. . 3 ⊢ ∃𝑥⊤
3	2	biantrur 530	. 2 ⊢ (∃*𝑥⊤ ↔ (∃𝑥⊤ ∧ ∃*𝑥⊤))
4		wl-moae 37470	. 2 ⊢ (∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
5	1, 3, 4	3bitr2i 299	1 ⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)

Syntax hints:   ↔ wb 206   ∧ wa 395  ∀wal 1535  ⊤wtru 1538  ∃wex 1777  ∃*wmo 2541  ∃!weu 2571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-mo 2543  df-eu 2572

This theorem is referenced by: (None)