Theorem mo4 2566

Description: At-most-one quantifier expressed using implicit substitution. This theorem is also a direct consequence of mo4f 2567, 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 2163. (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 2540	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧))
2		mo4.1	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3		equequ1 2027	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
4	2, 3	imbi12d 344	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) ↔ (𝜓 → 𝑦 = 𝑧)))
5	4	cbvalvw 2038	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∀𝑦(𝜓 → 𝑦 = 𝑧))
6	5	biimpi 216	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → ∀𝑦(𝜓 → 𝑦 = 𝑧))
7		pm2.27 42	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝜑 → 𝑥 = 𝑧) → 𝑥 = 𝑧))
8		pm2.27 42	. . . . . . . . . . 11 ⊢ (𝜓 → ((𝜓 → 𝑦 = 𝑧) → 𝑦 = 𝑧))
9	7, 8	im2anan9 621	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (((𝜑 → 𝑥 = 𝑧) ∧ (𝜓 → 𝑦 = 𝑧)) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑧)))
10		equtr2 2029	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦)
11	9, 10	syl6com 37	. . . . . . . . 9 ⊢ (((𝜑 → 𝑥 = 𝑧) ∧ (𝜓 → 𝑦 = 𝑧)) → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
12	11	ex 412	. . . . . . . 8 ⊢ ((𝜑 → 𝑥 = 𝑧) → ((𝜓 → 𝑦 = 𝑧) → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
13	12	alimdv 1918	. . . . . . 7 ⊢ ((𝜑 → 𝑥 = 𝑧) → (∀𝑦(𝜓 → 𝑦 = 𝑧) → ∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
14	13	com12 32	. . . . . 6 ⊢ (∀𝑦(𝜓 → 𝑦 = 𝑧) → ((𝜑 → 𝑥 = 𝑧) → ∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
15	14	alimdv 1918	. . . . 5 ⊢ (∀𝑦(𝜓 → 𝑦 = 𝑧) → (∀𝑥(𝜑 → 𝑥 = 𝑧) → ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
16	6, 15	mpcom 38	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
17	16	exlimiv 1932	. . 3 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) → ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
18	1, 17	sylbi 217	. 2 ⊢ (∃*𝑥𝜑 → ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
19	2	cbvexvw 2039	. . . . 5 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
20	19	biimpri 228	. . . 4 ⊢ (∃𝑦𝜓 → ∃𝑥𝜑)
21		ax6evr 2017	. . . . . . . 8 ⊢ ∃𝑧 𝑥 = 𝑧
22		pm3.2 469	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
23	22	imim1d 82	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (𝜓 → 𝑥 = 𝑦)))
24		ax7 2018	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
25	23, 24	syl8 76	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (𝜓 → (𝑥 = 𝑧 → 𝑦 = 𝑧))))
26	25	com4r 94	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝜑 → (((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (𝜓 → 𝑦 = 𝑧))))
27	26	impcom 407	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (𝜓 → 𝑦 = 𝑧)))
28	27	alimdv 1918	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → ∀𝑦(𝜓 → 𝑦 = 𝑧)))
29	28	impancom 451	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) → (𝑥 = 𝑧 → ∀𝑦(𝜓 → 𝑦 = 𝑧)))
30	29	eximdv 1919	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) → (∃𝑧 𝑥 = 𝑧 → ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)))
31	21, 30	mpi 20	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) → ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
32		dfmo 2540	. . . . . . 7 ⊢ (∃*𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
33	31, 32	sylibr 234	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) → ∃*𝑦𝜓)
34	33	expcom 413	. . . . 5 ⊢ (∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (𝜑 → ∃*𝑦𝜓))
35	34	aleximi 1834	. . . 4 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (∃𝑥𝜑 → ∃𝑥∃*𝑦𝜓))
36		ax5e 1914	. . . 4 ⊢ (∃𝑥∃*𝑦𝜓 → ∃*𝑦𝜓)
37	20, 35, 36	syl56 36	. . 3 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → (∃𝑦𝜓 → ∃*𝑦𝜓))
38	5	exbii 1850	. . . . 5 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
39	38, 1, 32	3bitr4i 303	. . . 4 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
40		moabs 2543	. . . 4 ⊢ (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃*𝑦𝜓))
41	39, 40	bitri 275	. . 3 ⊢ (∃*𝑥𝜑 ↔ (∃𝑦𝜓 → ∃*𝑦𝜓))
42	37, 41	sylibr 234	. 2 ⊢ (∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) → ∃*𝑥𝜑)
43	18, 42	impbii 209	1 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540  ∃wex 1781  ∃*wmo 2537

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 1912  ax-6 1969  ax-7 2010

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-mo 2539

This theorem is referenced by:  eu4  2615  moel  3362  moeq  3653  rmo4  3676  mosneq  4785  dffun6  6509  fun11  6572  brprcneu  6830  brprcneuALT  6831  dff13  7209  caovmo  7604  wemoiso  7926  wemoiso2  7927  addsrmo  10996  mulsrmo  10997  summo  15679  prodmo  15901  hausflimi  23945  vitalilem3  25577  plyexmo  26279  nosupprefixmo  27664  noinfprefixmo  27665  tglineintmo  28710  ajmoi  30929  pjhthmo  31373  adjmo  31903  satfv0  35540  satfv0fun  35553  satffunlem1lem1  35584  satffunlem2lem1  35586  funtransport  36213  funray  36322  funline  36324  lineintmo  36339  mopre  38792  cossssid4  38881  dffrege115  44405  mof0ALT  49315  mofsn  49319  f1omoOLD  49369  thincmo  49903  euendfunc  50001