Theorem mo3 2628

Description: Alternate definition of the at-most-one quantifier. Definition of [BellMachover] p. 460, except that definition has the side condition that 𝑦 not occur in 𝜑 in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Aug-2019.)

Hypothesis

Ref	Expression
mo3.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
mo3	⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem mo3

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfmo1 2570	. . 3 ⊢ Ⅎ𝑥∃*𝑥𝜑
2		mo3.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
3	2	nfmo 2579	. . . 4 ⊢ Ⅎ𝑦∃*𝑥𝜑
4		df-mo 2565	. . . . 5 ⊢ (∃*𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧))
5		sp 2215	. . . . . . . 8 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (𝜑 → 𝑥 = 𝑧))
6		spsbim 2485	. . . . . . . . 9 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝑥 = 𝑧))
7		equsb3 2524	. . . . . . . . 9 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧)
8	6, 7	syl6ib 242	. . . . . . . 8 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → ([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧))
9	5, 8	anim12d 602	. . . . . . 7 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑧)))
10		equtr2 2124	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦)
11	9, 10	syl6 35	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
12	11	exlimiv 2025	. . . . 5 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
13	4, 12	sylbi 208	. . . 4 ⊢ (∃*𝑥𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
14	3, 13	alrimi 2246	. . 3 ⊢ (∃*𝑥𝜑 → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
15	1, 14	alrimi 2246	. 2 ⊢ (∃*𝑥𝜑 → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
16		nfs1v 2287	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
17		pm3.21 463	. . . . . . . . 9 ⊢ ([𝑦 / 𝑥]𝜑 → (𝜑 → (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
18	17	imim1d 82	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝜑 → (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦)))
19	16, 18	alimd 2245	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 → (∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∀𝑥(𝜑 → 𝑥 = 𝑦)))
20	19	com12 32	. . . . . 6 ⊢ (∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑦)))
21	20	aleximi 1926	. . . . 5 ⊢ (∀𝑦∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (∃𝑦[𝑦 / 𝑥]𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
22	2	sb8e 2516	. . . . 5 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
23	2	mof 2571	. . . . 5 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
24	21, 22, 23	3imtr4g 287	. . . 4 ⊢ (∀𝑦∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (∃𝑥𝜑 → ∃*𝑥𝜑))
25		moabs 2575	. . . 4 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
26	24, 25	sylibr 225	. . 3 ⊢ (∀𝑦∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃*𝑥𝜑)
27	26	alcoms 2199	. 2 ⊢ (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃*𝑥𝜑)
28	15, 27	impbii 200	1 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384  ∀wal 1650  ∃wex 1874  Ⅎwnf 1878  [wsb 2062  ∃*wmo 2563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565

This theorem is referenced by:  mo  2629  eu2  2630  mo4f  2637  2mo  2673  rmo3f  3562  rmo3  3686  isarep2  6156  mo5f  29780  bnj580  31431  pm14.12  39295