Theorem mo3 2567

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.) Remove dependency on ax-13 2380. (Revised by BJ and WL, 29-Jan-2023.)

Hypothesis

Ref	Expression
mo3.nf	⊢ 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 2560	. . 3 ⊢ Ⅎ𝑥∃*𝑥𝜑
2		mo3.nf	. . . . 5 ⊢ Ⅎ𝑦𝜑
3	2	nfmov 2563	. . . 4 ⊢ Ⅎ𝑦∃*𝑥𝜑
4		df-mo 2543	. . . . 5 ⊢ (∃*𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧))
5		sp 2184	. . . . . . . 8 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (𝜑 → 𝑥 = 𝑧))
6		spsbim 2072	. . . . . . . . 9 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝑥 = 𝑧))
7		equsb3 2103	. . . . . . . . 9 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧)
8	6, 7	imbitrdi 251	. . . . . . . 8 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → ([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧))
9	5, 8	anim12d 608	. . . . . . 7 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑧)))
10		equtr2 2026	. . . . . . 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 1929	. . . . 5 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
13	4, 12	sylbi 217	. . . 4 ⊢ (∃*𝑥𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
14	3, 13	alrimi 2214	. . 3 ⊢ (∃*𝑥𝜑 → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
15	1, 14	alrimi 2214	. 2 ⊢ (∃*𝑥𝜑 → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
16		nfs1v 2157	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
17		pm3.21 471	. . . . . . . . 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 2213	. . . . . . 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 1830	. . . . 5 ⊢ (∀𝑦∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (∃𝑦[𝑦 / 𝑥]𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
22	2	sb8ef 2361	. . . . 5 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
23	2	mof 2566	. . . . 5 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
24	21, 22, 23	3imtr4g 296	. . . 4 ⊢ (∀𝑦∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (∃𝑥𝜑 → ∃*𝑥𝜑))
25		moabs 2546	. . . 4 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
26	24, 25	sylibr 234	. . 3 ⊢ (∀𝑦∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃*𝑥𝜑)
27	26	alcoms 2159	. 2 ⊢ (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃*𝑥𝜑)
28	15, 27	impbii 209	1 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535  ∃wex 1777  Ⅎwnf 1781  [wsb 2064  ∃*wmo 2541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543

This theorem is referenced by:  mo  2568  mo4f  2570  eu2  2612  2mo  2651  rmo3f  3756  rmo3  3911  isarep2  6669  mo5f  32517  bnj580  34889  pm14.12  44390