Theorem wl-euae 37980

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 2595	. 2 ⊢ (∃!𝑥⊤ ↔ (∃𝑥⊤ ∧ ∃*𝑥⊤))
2		extru 1994	. . 3 ⊢ ∃𝑥⊤
3	2	biantrur 538	. 2 ⊢ (∃*𝑥⊤ ↔ (∃𝑥⊤ ∧ ∃*𝑥⊤))
4		wl-moae 37979	. 2 ⊢ (∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
5	1, 3, 4	3bitr2i 301	1 ⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)

Syntax hints:   ↔ wb 208   ∧ wa 399  ∀wal 1557  ⊤wtru 1560  ∃wex 1798  ∃*wmo 2563  ∃!weu 2594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-mo 2565  df-eu 2595

This theorem is referenced by: (None)