Theorem mo4 2561

Description: At-most-one quantifier expressed using implicit substitution. This theorem is also a direct consequence of mo4f 2562, 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 2155. (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 2535	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧))
2		mo4.1	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3		equequ1 2029	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
4	2, 3	imbi12d 345	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) ↔ (𝜓 → 𝑦 = 𝑧)))
5	4	cbvalvw 2040	. . . . . 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 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → ((𝜑 → 𝑥 = 𝑧) → 𝑥 = 𝑧))
9		pm2.27 42	. . . . . . . . . . . 12 ⊢ (𝜓 → ((𝜓 → 𝑦 = 𝑧) → 𝑦 = 𝑧))
10	9	adantl 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → ((𝜓 → 𝑦 = 𝑧) → 𝑦 = 𝑧))
11	8, 10	anim12d 610	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (((𝜑 → 𝑥 = 𝑧) ∧ (𝜓 → 𝑦 = 𝑧)) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑧)))
12		equtr2 2031	. . . . . . . . . 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 414	. . . . . . . 8 ⊢ ((𝜑 → 𝑥 = 𝑧) → ((𝜓 → 𝑦 = 𝑧) → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
15	14	alimdv 1920	. . . . . . 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 1920	. . . . 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 1934	. . 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 2041	. . . . 5 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
22	21	biimpri 227	. . . 4 ⊢ (∃𝑦𝜓 → ∃𝑥𝜑)
23		ax6evr 2019	. . . . . . . 8 ⊢ ∃𝑧 𝑥 = 𝑧
24		pm3.2 471	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
25	24	imim1d 82	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (𝜓 → 𝑥 = 𝑦)))
26		ax7 2020	. . . . . . . . . . . . . 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 409	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (𝜓 → 𝑦 = 𝑧)))
30	29	alimdv 1920	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → ∀𝑦(𝜓 → 𝑦 = 𝑧)))
31	30	impancom 453	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) → (𝑥 = 𝑧 → ∀𝑦(𝜓 → 𝑦 = 𝑧)))
32	31	eximdv 1921	. . . . . . . 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 2535	. . . . . . 7 ⊢ (∃*𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
35	33, 34	sylibr 233	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) → ∃*𝑦𝜓)
36	35	expcom 415	. . . . 5 ⊢ (∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (𝜑 → ∃*𝑦𝜓))
37	36	aleximi 1835	. . . 4 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (∃𝑥𝜑 → ∃𝑥∃*𝑦𝜓))
38		ax5e 1916	. . . 4 ⊢ (∃𝑥∃*𝑦𝜓 → ∃*𝑦𝜓)
39	22, 37, 38	syl56 36	. . 3 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (∃𝑦𝜓 → ∃*𝑦𝜓))
40	5	exbii 1851	. . . . 5 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
41	40, 1, 34	3bitr4i 303	. . . 4 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
42		moabs 2538	. . . 4 ⊢ (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃*𝑦𝜓))
43	41, 42	bitri 275	. . 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 397  ∀wal 1540  ∃wex 1782  ∃*wmo 2533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-mo 2535

This theorem is referenced by:  eu4  2612  moel  3399  moelOLD  3402  moeq  3704  rmo4  3727  mosneq  4844  dffun6  6557  dffun3OLD  6559  fun11  6623  brprcneu  6882  brprcneuALT  6883  dff13  7254  mpofunOLD  7533  caovmo  7644  wemoiso  7960  wemoiso2  7961  addsrmo  11068  mulsrmo  11069  summo  15663  prodmo  15880  hausflimi  23484  vitalilem3  25127  plyexmo  25826  nosupprefixmo  27203  noinfprefixmo  27204  tglineintmo  27893  ajmoi  30111  pjhthmo  30555  adjmo  31085  satfv0  34349  satfv0fun  34362  satffunlem1lem1  34393  satffunlem2lem1  34395  funtransport  35003  funray  35112  funline  35114  lineintmo  35129  cossssid4  37340  dffrege115  42729  mof0ALT  47506  mofsn  47510  f1omo  47527  thincmo  47649