Theorem 2eu4 2679

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 2675 for a condition under which the naive definition holds and 2exeu 2671 for a one-way implication. See 2eu5 2680 and 2eu8 2683 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 2594	. . 3 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑))
2		df-eu 2594	. . . 4 ⊢ (∃!𝑦∃𝑥𝜑 ↔ (∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑))
3		excom 2198	. . . 4 ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑)
4	2, 3	bianbi 639	. . 3 ⊢ (∃!𝑦∃𝑥𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑))
5	1, 4	anbi12i 640	. 2 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) ∧ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑)))
6		anandi 689	. 2 ⊢ ((∃𝑥∃𝑦𝜑 ∧ (∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑)) ↔ ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) ∧ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑)))
7		2mo2 2672	. . 3 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑) ↔ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
8	7	anbi2i 635	. 2 ⊢ ((∃𝑥∃𝑦𝜑 ∧ (∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑)) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
9	5, 6, 8	3bitr2i 302	1 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567  ∃wex 1811  ∃*wmo 2562  ∃!weu 2593

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-11 2193

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-mo 2564  df-eu 2594

This theorem is used by:  2eu5  2680  2eu6  2681