Theorem mo4 2594

Description: At-most-one quantifier expressed using implicit substitution. This theorem is also a direct consequence of mo4f 2595, but this proof is based on fewer axioms.

Note: Swapping both 𝑥, 𝑦 and 𝜑, 𝜓 preserves the hypothesis and DV conditions, so we also have the conclusion ⊢ (∃*𝑦𝜓 ↔ ∀𝑦∀𝑥((𝜓 ∧ 𝜑) → 𝑦 = 𝑥)). But under the hypothesis and DV condition, we also have ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) (that is a proof step). Therefore, since ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ↔ 𝜑)) (ancom 466) and ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) (equcom 2052) , we obtain, still under the hypothesis and DV condition, ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ ∀𝑦∀𝑥((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)); in other words, in this special case we can swap universal quantifiers as in alcom 2198 without requiring ax-11 2196. (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		dfmo 2568	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧))
2		mo4.1	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3		equequ1 2059	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
4	2, 3	imbi12d 347	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) ↔ (𝜓 → 𝑦 = 𝑧)))
5	4	cbvalvw 2070	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∀𝑦(𝜓 → 𝑦 = 𝑧))
6	5	biimpi 219	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → ∀𝑦(𝜓 → 𝑦 = 𝑧))
7		pm2.27 43	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝜑 → 𝑥 = 𝑧) → 𝑥 = 𝑧))
8		pm2.27 43	. . . . . . . . . . 11 ⊢ (𝜓 → ((𝜓 → 𝑦 = 𝑧) → 𝑦 = 𝑧))
9	7, 8	im2anan9 632	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (((𝜑 → 𝑥 = 𝑧) ∧ (𝜓 → 𝑦 = 𝑧)) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑧)))
10		equtr2 2061	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦)
11	9, 10	syl6com 38	. . . . . . . . 9 ⊢ (((𝜑 → 𝑥 = 𝑧) ∧ (𝜓 → 𝑦 = 𝑧)) → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
12	11	ex 418	. . . . . . . 8 ⊢ ((𝜑 → 𝑥 = 𝑧) → ((𝜓 → 𝑦 = 𝑧) → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
13	12	alimdv 1950	. . . . . . 7 ⊢ ((𝜑 → 𝑥 = 𝑧) → (∀𝑦(𝜓 → 𝑦 = 𝑧) → ∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
14	13	com12 33	. . . . . 6 ⊢ (∀𝑦(𝜓 → 𝑦 = 𝑧) → ((𝜑 → 𝑥 = 𝑧) → ∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
15	14	alimdv 1950	. . . . 5 ⊢ (∀𝑦(𝜓 → 𝑦 = 𝑧) → (∀𝑥(𝜑 → 𝑥 = 𝑧) → ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
16	6, 15	mpcom 39	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
17	16	exlimiv 1964	. . 3 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) → ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
18	1, 17	sylbi 220	. 2 ⊢ (∃*𝑥𝜑 → ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
19	2	cbvexvw 2071	. . . . 5 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
20	19	biimpri 231	. . . 4 ⊢ (∃𝑦𝜓 → ∃𝑥𝜑)
21		ax6evr 2049	. . . . . . . 8 ⊢ ∃𝑧 𝑥 = 𝑧
22		pm3.2 475	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
23	22	imim1d 83	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (𝜓 → 𝑥 = 𝑦)))
24		ax7 2050	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
25	23, 24	syl8 77	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (𝜓 → (𝑥 = 𝑧 → 𝑦 = 𝑧))))
26	25	com4r 95	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝜑 → (((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (𝜓 → 𝑦 = 𝑧))))
27	26	impcom 413	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (𝜓 → 𝑦 = 𝑧)))
28	27	alimdv 1950	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → ∀𝑦(𝜓 → 𝑦 = 𝑧)))
29	28	impancom 457	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) → (𝑥 = 𝑧 → ∀𝑦(𝜓 → 𝑦 = 𝑧)))
30	29	eximdv 1951	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) → (∃𝑧 𝑥 = 𝑧 → ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)))
31	21, 30	mpi 21	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) → ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
32		dfmo 2568	. . . . . . 7 ⊢ (∃*𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
33	31, 32	sylibr 237	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) → ∃*𝑦𝜓)
34	33	expcom 419	. . . . 5 ⊢ (∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (𝜑 → ∃*𝑦𝜓))
35	34	aleximi 1864	. . . 4 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (∃𝑥𝜑 → ∃𝑥∃*𝑦𝜓))
36		ax5e 1946	. . . 4 ⊢ (∃𝑥∃*𝑦𝜓 → ∃*𝑦𝜓)
37	20, 35, 36	syl56 37	. . 3 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (∃𝑦𝜓 → ∃*𝑦𝜓))
38	5	exbii 1880	. . . . 5 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
39	38, 1, 32	3bitr4i 306	. . . 4 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
40		moabs 2571	. . . 4 ⊢ (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃*𝑦𝜓))
41	39, 40	bitri 278	. . 3 ⊢ (∃*𝑥𝜑 ↔ (∃𝑦𝜓 → ∃*𝑦𝜓))
42	37, 41	sylibr 237	. 2 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → ∃*𝑥𝜑)
43	18, 42	impbii 212	1 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567  ∃wex 1811  ∃*wmo 2565

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-mo 2567

This theorem is used by:  eu4  2643  moel  3389  moeq  3668  rmo4  3691  mosneq  4800  dffun6  6558  fun11  6625  brprcneu  6886  brprcneuALT  6887  dff13  7268  caovmo  7668  wemoiso  7995  wemoiso2  7996  addsrmo  11171  mulsrmo  11172  summo  15898  prodmo  16118  hausflimi  24314  vitalilem3  25946  plyexmo  26651  nosupprefixmo  28072  noinfprefixmo  28073  tglineintmo  29125  ajmoi  31475  pjhthmo  31919  adjmo  32449  satfv0  36247  satfv0fun  36260  satffunlem1lem1  36291  satffunlem2lem1  36293  funtransport  36926  funray  37035  funline  37037  lineintmo  37052  mopre  39546  cossssid4  39635  dffrege115  45121  mof0ALT  50093  mofsn  50097  f1omoOLD  50150  thincmo  50705  euendfunc  50803