Theorem mo4 2564

Description: At-most-one quantifier expressed using implicit substitution. This theorem is also a direct consequence of mo4f 2565, but this proof is based on fewer axioms.

By the way, swapping 𝑥, 𝑦 and 𝜑, 𝜓 leads to an expression for ∃*𝑦𝜓, which is equivalent to ∃*𝑥𝜑 (is a proof line), so the right hand side is a rare instance of an expression where swapping the quantifiers can be done without ax-11 2154. (Contributed by NM, 26-Jul-1995.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Oct-2023.)

Hypothesis

Ref	Expression
mo4.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
mo4	⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem mo4

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mo 2538	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧))
2		mo4.1	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3		equequ1 2028	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
4	2, 3	imbi12d 344	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) ↔ (𝜓 → 𝑦 = 𝑧)))
5	4	cbvalvw 2039	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∀𝑦(𝜓 → 𝑦 = 𝑧))
6	5	biimpi 215	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → ∀𝑦(𝜓 → 𝑦 = 𝑧))
7		pm2.27 42	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝜑 → 𝑥 = 𝑧) → 𝑥 = 𝑧))
8	7	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → ((𝜑 → 𝑥 = 𝑧) → 𝑥 = 𝑧))
9		pm2.27 42	. . . . . . . . . . . 12 ⊢ (𝜓 → ((𝜓 → 𝑦 = 𝑧) → 𝑦 = 𝑧))
10	9	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → ((𝜓 → 𝑦 = 𝑧) → 𝑦 = 𝑧))
11	8, 10	anim12d 609	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (((𝜑 → 𝑥 = 𝑧) ∧ (𝜓 → 𝑦 = 𝑧)) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑧)))
12		equtr2 2030	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦)
13	11, 12	syl6com 37	. . . . . . . . 9 ⊢ (((𝜑 → 𝑥 = 𝑧) ∧ (𝜓 → 𝑦 = 𝑧)) → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
14	13	ex 413	. . . . . . . 8 ⊢ ((𝜑 → 𝑥 = 𝑧) → ((𝜓 → 𝑦 = 𝑧) → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
15	14	alimdv 1919	. . . . . . 7 ⊢ ((𝜑 → 𝑥 = 𝑧) → (∀𝑦(𝜓 → 𝑦 = 𝑧) → ∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
16	15	com12 32	. . . . . 6 ⊢ (∀𝑦(𝜓 → 𝑦 = 𝑧) → ((𝜑 → 𝑥 = 𝑧) → ∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
17	16	alimdv 1919	. . . . 5 ⊢ (∀𝑦(𝜓 → 𝑦 = 𝑧) → (∀𝑥(𝜑 → 𝑥 = 𝑧) → ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
18	6, 17	mpcom 38	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
19	18	exlimiv 1933	. . 3 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) → ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
20	1, 19	sylbi 216	. 2 ⊢ (∃*𝑥𝜑 → ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
21	2	cbvexvw 2040	. . . . 5 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
22	21	biimpri 227	. . . 4 ⊢ (∃𝑦𝜓 → ∃𝑥𝜑)
23		ax6evr 2018	. . . . . . . 8 ⊢ ∃𝑧 𝑥 = 𝑧
24		pm3.2 470	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
25	24	imim1d 82	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (𝜓 → 𝑥 = 𝑦)))
26		ax7 2019	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
27	25, 26	syl8 76	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (𝜓 → (𝑥 = 𝑧 → 𝑦 = 𝑧))))
28	27	com4r 94	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝜑 → (((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (𝜓 → 𝑦 = 𝑧))))
29	28	impcom 408	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (𝜓 → 𝑦 = 𝑧)))
30	29	alimdv 1919	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → ∀𝑦(𝜓 → 𝑦 = 𝑧)))
31	30	impancom 452	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) → (𝑥 = 𝑧 → ∀𝑦(𝜓 → 𝑦 = 𝑧)))
32	31	eximdv 1920	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) → (∃𝑧 𝑥 = 𝑧 → ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)))
33	23, 32	mpi 20	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) → ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
34		df-mo 2538	. . . . . . 7 ⊢ (∃*𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
35	33, 34	sylibr 233	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) → ∃*𝑦𝜓)
36	35	expcom 414	. . . . 5 ⊢ (∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (𝜑 → ∃*𝑦𝜓))
37	36	aleximi 1834	. . . 4 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (∃𝑥𝜑 → ∃𝑥∃*𝑦𝜓))
38		ax5e 1915	. . . 4 ⊢ (∃𝑥∃*𝑦𝜓 → ∃*𝑦𝜓)
39	22, 37, 38	syl56 36	. . 3 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (∃𝑦𝜓 → ∃*𝑦𝜓))
40	5	exbii 1850	. . . . 5 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
41	40, 1, 34	3bitr4i 302	. . . 4 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
42		moabs 2541	. . . 4 ⊢ (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃*𝑦𝜓))
43	41, 42	bitri 274	. . 3 ⊢ (∃*𝑥𝜑 ↔ (∃𝑦𝜓 → ∃*𝑦𝜓))
44	39, 43	sylibr 233	. 2 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → ∃*𝑥𝜑)
45	20, 44	impbii 208	1 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1539  ∃wex 1781  ∃*wmo 2536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-mo 2538

This theorem is referenced by:  eu4  2615  moel  3375  moelOLD  3378  moeq  3665  rmo4  3688  mosneq  4800  dffun6  6509  dffun3OLD  6511  fun11  6575  brprcneu  6832  brprcneuALT  6833  dff13  7202  mpofunOLD  7481  caovmo  7591  wemoiso  7906  wemoiso2  7907  addsrmo  11009  mulsrmo  11010  summo  15602  prodmo  15819  hausflimi  23331  vitalilem3  24974  plyexmo  25673  nosupprefixmo  27048  noinfprefixmo  27049  tglineintmo  27584  ajmoi  29800  pjhthmo  30244  adjmo  30774  satfv0  33952  satfv0fun  33965  satffunlem1lem1  33996  satffunlem2lem1  33998  funtransport  34616  funray  34725  funline  34727  lineintmo  34742  cossssid4  36932  dffrege115  42240  mof0ALT  46896  mofsn  46900  f1omo  46917  thincmo  47039