Theorem wl-mo3t 37577

Description: Closed form of mo3 2564. (Contributed by Wolf Lammen, 18-Aug-2019.)

Assertion

Ref	Expression
wl-mo3t	⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem wl-mo3t

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfa1 2151	. . 3 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑
2		nfmo1 2557	. . 3 ⊢ Ⅎ𝑥∃*𝑥𝜑
3		nfnf1 2154	. . . . . . 7 ⊢ Ⅎ𝑦Ⅎ𝑦𝜑
4	3	nfal 2323	. . . . . 6 ⊢ Ⅎ𝑦∀𝑥Ⅎ𝑦𝜑
5		sp 2183	. . . . . . 7 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦𝜑)
6	1, 5	nfmodv 2559	. . . . . 6 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∃*𝑥𝜑)
7	4, 6	nfan1 2200	. . . . 5 ⊢ Ⅎ𝑦(∀𝑥Ⅎ𝑦𝜑 ∧ ∃*𝑥𝜑)
8		df-mo 2540	. . . . . . 7 ⊢ (∃*𝑥𝜑 ↔ ∃𝑢∀𝑥(𝜑 → 𝑥 = 𝑢))
9		sp 2183	. . . . . . . . . 10 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑢) → (𝜑 → 𝑥 = 𝑢))
10		spsbim 2072	. . . . . . . . . . 11 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑢) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝑥 = 𝑢))
11		equsb3 2103	. . . . . . . . . . 11 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑢 ↔ 𝑦 = 𝑢)
12	10, 11	imbitrdi 251	. . . . . . . . . 10 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑢) → ([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑢))
13	9, 12	anim12d 609	. . . . . . . . 9 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑢) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝑥 = 𝑢 ∧ 𝑦 = 𝑢)))
14		equtr2 2026	. . . . . . . . 9 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑢) → 𝑥 = 𝑦)
15	13, 14	syl6 35	. . . . . . . 8 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑢) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
16	15	exlimiv 1930	. . . . . . 7 ⊢ (∃𝑢∀𝑥(𝜑 → 𝑥 = 𝑢) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
17	8, 16	sylbi 217	. . . . . 6 ⊢ (∃*𝑥𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
18	17	adantl 481	. . . . 5 ⊢ ((∀𝑥Ⅎ𝑦𝜑 ∧ ∃*𝑥𝜑) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
19	7, 18	alrimi 2213	. . . 4 ⊢ ((∀𝑥Ⅎ𝑦𝜑 ∧ ∃*𝑥𝜑) → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
20	19	ex 412	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
21	1, 2, 20	alrimd 2215	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
22		nfa1 2151	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)
23		nfs1v 2156	. . . . . 6 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
24		pm3.3 448	. . . . . . . 8 ⊢ (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (𝜑 → ([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
25	24	com23 86	. . . . . . 7 ⊢ (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 → (𝜑 → 𝑥 = 𝑦)))
26	25	sps 2185	. . . . . 6 ⊢ (∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 → (𝜑 → 𝑥 = 𝑦)))
27	22, 23, 26	alrimd 2215	. . . . 5 ⊢ (∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑦)))
28	27	aleximi 1832	. . . 4 ⊢ (∀𝑦∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (∃𝑦[𝑦 / 𝑥]𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
29	28	alcoms 2158	. . 3 ⊢ (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (∃𝑦[𝑦 / 𝑥]𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
30		moabs 2543	. . . 4 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
31		wl-sb8eft 37552	. . . . 5 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))
32		wl-mo2t 37576	. . . . 5 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
33	31, 32	imbi12d 344	. . . 4 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ((∃𝑥𝜑 → ∃*𝑥𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))))
34	30, 33	bitrid 283	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))))
35	29, 34	imbitrrid 246	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃*𝑥𝜑))
36	21, 35	impbid 212	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538  ∃wex 1779  Ⅎwnf 1783  [wsb 2064  ∃*wmo 2538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2157  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540

This theorem is referenced by: (None)