Theorem 2mo2 2538

Description: This theorem extends the idea of "at most one" to expressions in two set variables ("at most one pair 𝑥 and 𝑦". Note: this is not expressed by ∃*𝑥∃*𝑦𝜑). 2eu4 2544 relates this extension to double existential uniqueness, if at least one pair exists. (Contributed by Wolf Lammen, 26-Oct-2019.)

Assertion

Ref	Expression
2mo2	⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑) ↔ ∃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 2mo2

Step	Hyp	Ref	Expression
1		eeanv 2170	. 2 ⊢ (∃𝑧∃𝑤(∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤)) ↔ (∃𝑧∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧) ∧ ∃𝑤∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤)))
2		jcab 903	. . . . 5 ⊢ ((𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ((𝜑 → 𝑥 = 𝑧) ∧ (𝜑 → 𝑦 = 𝑤)))
3	2	2albii 1738	. . . 4 ⊢ (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥∀𝑦((𝜑 → 𝑥 = 𝑧) ∧ (𝜑 → 𝑦 = 𝑤)))
4		19.26-2 1787	. . . 4 ⊢ (∀𝑥∀𝑦((𝜑 → 𝑥 = 𝑧) ∧ (𝜑 → 𝑦 = 𝑤)) ↔ (∀𝑥∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤)))
5		19.23v 1889	. . . . . 6 ⊢ (∀𝑦(𝜑 → 𝑥 = 𝑧) ↔ (∃𝑦𝜑 → 𝑥 = 𝑧))
6	5	albii 1737	. . . . 5 ⊢ (∀𝑥∀𝑦(𝜑 → 𝑥 = 𝑧) ↔ ∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧))
7		alcom 2024	. . . . . 6 ⊢ (∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤) ↔ ∀𝑦∀𝑥(𝜑 → 𝑦 = 𝑤))
8		19.23v 1889	. . . . . . 7 ⊢ (∀𝑥(𝜑 → 𝑦 = 𝑤) ↔ (∃𝑥𝜑 → 𝑦 = 𝑤))
9	8	albii 1737	. . . . . 6 ⊢ (∀𝑦∀𝑥(𝜑 → 𝑦 = 𝑤) ↔ ∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤))
10	7, 9	bitri 263	. . . . 5 ⊢ (∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤) ↔ ∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤))
11	6, 10	anbi12i 729	. . . 4 ⊢ ((∀𝑥∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤)) ↔ (∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤)))
12	3, 4, 11	3bitri 285	. . 3 ⊢ (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤)))
13	12	2exbii 1765	. 2 ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧∃𝑤(∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤)))
14		mo2v 2465	. . 3 ⊢ (∃*𝑥∃𝑦𝜑 ↔ ∃𝑧∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧))
15		mo2v 2465	. . 3 ⊢ (∃*𝑦∃𝑥𝜑 ↔ ∃𝑤∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤))
16	14, 15	anbi12i 729	. 2 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑) ↔ (∃𝑧∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧) ∧ ∃𝑤∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤)))
17	1, 13, 16	3bitr4ri 292	1 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑) ↔ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695  ∃*wmo 2459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463

This theorem is referenced by:  2mo  2539  2eu4  2544