Theorem 2eu5 2288

Description: An alternate definition of double existential uniqueness (see 2eu4 2287). A mistake sometimes made in the literature is to use ∃!x∃!y to mean "exactly one x and exactly one y." (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining ∀x∃*yφ as an additional condition. The correct definition apparently has never been published. (∃* means "there exists at most one".) (Contributed by NM, 26-Oct-2003.)

Assertion

Ref	Expression
2eu5	⊢ ((∃!x∃!yφ ∧ ∀x∃*yφ) ↔ (∃x∃yφ ∧ ∃z∃w∀x∀y(φ → (x = z ∧ y = w))))

Distinct variable groups:   x,y,z,w   φ,z,w

Allowed substitution hints:   φ(x,y)

Proof of Theorem 2eu5

Step	Hyp	Ref	Expression
1		2eu1 2284	. . 3 ⊢ (∀x∃*yφ → (∃!x∃!yφ ↔ (∃!x∃yφ ∧ ∃!y∃xφ)))
2	1	pm5.32ri 619	. 2 ⊢ ((∃!x∃!yφ ∧ ∀x∃*yφ) ↔ ((∃!x∃yφ ∧ ∃!y∃xφ) ∧ ∀x∃*yφ))
3		eumo 2244	. . . . 5 ⊢ (∃!y∃xφ → ∃*y∃xφ)
4	3	adantl 452	. . . 4 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) → ∃*y∃xφ)
5		2moex 2275	. . . 4 ⊢ (∃*y∃xφ → ∀x∃*yφ)
6	4, 5	syl 15	. . 3 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) → ∀x∃*yφ)
7	6	pm4.71i 613	. 2 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) ↔ ((∃!x∃yφ ∧ ∃!y∃xφ) ∧ ∀x∃*yφ))
8		2eu4 2287	. 2 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) ↔ (∃x∃yφ ∧ ∃z∃w∀x∀y(φ → (x = z ∧ y = w))))
9	2, 7, 8	3bitr2i 264	1 ⊢ ((∃!x∃!yφ ∧ ∀x∃*yφ) ↔ (∃x∃yφ ∧ ∃z∃w∀x∀y(φ → (x = z ∧ y = w))))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  ∃!weu 2204  ∃*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by:  2reu5lem3  3044