Theorem dfmoeu 2534

Description: An elementary proof of moeu 2581 in disguise, connecting an expression characterizing uniqueness (df-mo 2538) to that of existential uniqueness (eu6 2572). No particular order of definition is required, as one can be derived from the other. This is shown here and in dfeumo 2535. (Contributed by Wolf Lammen, 27-May-2019.)

Assertion

Ref	Expression
dfmoeu	⊢ ((∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem dfmoeu

Step	Hyp	Ref	Expression
1		alnex 1781	. . . . 5 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2		pm2.21 123	. . . . . 6 ⊢ (¬ 𝜑 → (𝜑 → 𝑥 = 𝑦))
3	2	alimi 1811	. . . . 5 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑦))
4	1, 3	sylbir 234	. . . 4 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑦))
5	4	19.8ad 2173	. . 3 ⊢ (¬ ∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
6		biimp 214	. . . . 5 ⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦))
7	6	alimi 1811	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥(𝜑 → 𝑥 = 𝑦))
8	7	eximi 1835	. . 3 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
9	5, 8	ja 186	. 2 ⊢ ((∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
10		nfia1 2148	. . . . 5 ⊢ Ⅎ𝑥(∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
11		id 22	. . . . . . . . 9 ⊢ (𝜑 → 𝜑)
12		ax12v 2170	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
13	12	com12 32	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
14	11, 13	embantd 59	. . . . . . . 8 ⊢ (𝜑 → ((𝜑 → 𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
15	14	spsd 2178	. . . . . . 7 ⊢ (𝜑 → (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
16	15	ancld 552	. . . . . 6 ⊢ (𝜑 → (∀𝑥(𝜑 → 𝑥 = 𝑦) → (∀𝑥(𝜑 → 𝑥 = 𝑦) ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑))))
17		albiim 1890	. . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (∀𝑥(𝜑 → 𝑥 = 𝑦) ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
18	16, 17	syl6ibr 252	. . . . 5 ⊢ (𝜑 → (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
19	10, 18	exlimi 2208	. . . 4 ⊢ (∃𝑥𝜑 → (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
20	19	eximdv 1918	. . 3 ⊢ (∃𝑥𝜑 → (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
21	20	com12 32	. 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
22	9, 21	impbii 208	1 ⊢ ((∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397  ∀wal 1537  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-10 2135  ax-12 2169

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-ex 1780  df-nf 1784

This theorem is referenced by:  dfeumo  2535  eu6  2572  dfmo  2594