Theorem 2eu4 2656

Description: This theorem provides us with a definition of double existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"). Naively one might think (incorrectly) that it could be defined by ∃!𝑥∃!𝑦𝜑. See 2eu1 2652 for a condition under which the naive definition holds and 2exeu 2648 for a one-way implication. See 2eu5 2657 and 2eu8 2660 for alternate definitions. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 14-Sep-2019.)

Assertion

Ref	Expression
2eu4	⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃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 2eu4

Step	Hyp	Ref	Expression
1		df-eu 2569	. . 3 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑))
2		df-eu 2569	. . . 4 ⊢ (∃!𝑦∃𝑥𝜑 ↔ (∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑))
3		excom 2162	. . . . 5 ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑)
4	3	anbi1i 624	. . . 4 ⊢ ((∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑))
5	2, 4	bitri 274	. . 3 ⊢ (∃!𝑦∃𝑥𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑))
6	1, 5	anbi12i 627	. 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) ∧ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑)))
7		anandi 673	. 2 ⊢ ((∃𝑥∃𝑦𝜑 ∧ (∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑)) ↔ ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) ∧ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑)))
8		2mo2 2649	. . 3 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑) ↔ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
9	8	anbi2i 623	. 2 ⊢ ((∃𝑥∃𝑦𝜑 ∧ (∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑)) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
10	6, 7, 9	3bitr2i 299	1 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537  ∃wex 1782  ∃*wmo 2538  ∃!weu 2568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-11 2154

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-mo 2540  df-eu 2569

This theorem is referenced by:  2eu5  2657  2eu6  2658