Theorem dfmoeu 2566

Description: An elementary proof of moeu 2614 in disguise, connecting an expression characterizing uniqueness (df-mo 2570) to that of existential uniqueness (eu6 2605). No particular order of definition is required, as one can be derived from the other. This is shown here and in dfeumo 2567. (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 1814	. . . . 5 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
2		pm2.21 124	. . . . . 6 ⊢ (¬ 𝜑 → (𝜑 → 𝑥 = 𝑦))
3	2	alimi 1844	. . . . 5 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑦))
4	1, 3	sylbir 238	. . . 4 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑦))
5	4	19.8ad 2221	. . 3 ⊢ (¬ ∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
6		biimp 218	. . . . 5 ⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦))
7	6	alimi 1844	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥(𝜑 → 𝑥 = 𝑦))
8	7	eximi 1868	. . 3 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
9	5, 8	ja 188	. 2 ⊢ ((∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
10		nfia1 2191	. . . . 5 ⊢ Ⅎ𝑥(∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
11		id 23	. . . . . . . . 9 ⊢ (𝜑 → 𝜑)
12		ax12v 2217	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
13	12	com12 33	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
14	11, 13	embantd 60	. . . . . . . 8 ⊢ (𝜑 → ((𝜑 → 𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
15	14	spsd 2226	. . . . . . 7 ⊢ (𝜑 → (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
16	15	ancld 560	. . . . . 6 ⊢ (𝜑 → (∀𝑥(𝜑 → 𝑥 = 𝑦) → (∀𝑥(𝜑 → 𝑥 = 𝑦) ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑))))
17		albiim 1922	. . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (∀𝑥(𝜑 → 𝑥 = 𝑦) ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
18	16, 17	imbitrrdi 255	. . . . 5 ⊢ (𝜑 → (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
19	10, 18	exlimi 2256	. . . 4 ⊢ (∃𝑥𝜑 → (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
20	19	eximdv 1950	. . 3 ⊢ (∃𝑥𝜑 → (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
21	20	com12 33	. 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
22	9, 21	impbii 212	1 ⊢ ((∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by:  dfeumo  2567  eu6  2605  dfmo2  2627