Theorem 2mo2 2649

Description: Two ways of expressing "there exists at most one ordered pair ⟨𝑥, 𝑦⟩ such that 𝜑(𝑥, 𝑦) holds. Note that this is not equivalent to ∃*𝑥∃*𝑦𝜑. See also 2mo 2650. This is the analogue of 2eu4 2656 for existential uniqueness. (Contributed by Wolf Lammen, 26-Oct-2019.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 3-Jan-2023.)

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		exdistrv 1960	. 2 ⊢ (∃𝑧∃𝑤(∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤)) ↔ (∃𝑧∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧) ∧ ∃𝑤∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤)))
2		jcab 517	. . . . 5 ⊢ ((𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ((𝜑 → 𝑥 = 𝑧) ∧ (𝜑 → 𝑦 = 𝑤)))
3	2	2albii 1824	. . . 4 ⊢ (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥∀𝑦((𝜑 → 𝑥 = 𝑧) ∧ (𝜑 → 𝑦 = 𝑤)))
4		19.26-2 1875	. . . 4 ⊢ (∀𝑥∀𝑦((𝜑 → 𝑥 = 𝑧) ∧ (𝜑 → 𝑦 = 𝑤)) ↔ (∀𝑥∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤)))
5		19.23v 1946	. . . . . 6 ⊢ (∀𝑦(𝜑 → 𝑥 = 𝑧) ↔ (∃𝑦𝜑 → 𝑥 = 𝑧))
6	5	albii 1823	. . . . 5 ⊢ (∀𝑥∀𝑦(𝜑 → 𝑥 = 𝑧) ↔ ∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧))
7		alcom 2158	. . . . . 6 ⊢ (∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤) ↔ ∀𝑦∀𝑥(𝜑 → 𝑦 = 𝑤))
8		19.23v 1946	. . . . . . 7 ⊢ (∀𝑥(𝜑 → 𝑦 = 𝑤) ↔ (∃𝑥𝜑 → 𝑦 = 𝑤))
9	8	albii 1823	. . . . . 6 ⊢ (∀𝑦∀𝑥(𝜑 → 𝑦 = 𝑤) ↔ ∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤))
10	7, 9	bitri 274	. . . . 5 ⊢ (∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤) ↔ ∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤))
11	6, 10	anbi12i 626	. . . 4 ⊢ ((∀𝑥∀𝑦(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑥∀𝑦(𝜑 → 𝑦 = 𝑤)) ↔ (∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤)))
12	3, 4, 11	3bitri 296	. . 3 ⊢ (∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤)))
13	12	2exbii 1852	. 2 ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧∃𝑤(∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤)))
14		df-mo 2540	. . 3 ⊢ (∃*𝑥∃𝑦𝜑 ↔ ∃𝑧∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧))
15		df-mo 2540	. . 3 ⊢ (∃*𝑦∃𝑥𝜑 ↔ ∃𝑤∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤))
16	14, 15	anbi12i 626	. 2 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑) ↔ (∃𝑧∀𝑥(∃𝑦𝜑 → 𝑥 = 𝑧) ∧ ∃𝑤∀𝑦(∃𝑥𝜑 → 𝑦 = 𝑤)))
17	1, 13, 16	3bitr4ri 303	1 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑) ↔ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537  ∃wex 1783  ∃*wmo 2538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-11 2156

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-mo 2540

This theorem is referenced by:  2mo  2650  2eu4  2656