Theorem rmo6 35756

Description: Alternate version of rmo2 3833 where 𝑦 has a restricted quantifier. The antecedent ∃𝑧𝑧 ∈ 𝐴 on the right-hand side handles the case where 𝐴 is empty. (Contributed by BTernaryTau, 29-Sep-2026.)

Hypothesis

Ref	Expression
rmo6.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
rmo6	⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑧 𝑧 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑧,𝐴

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)

Proof of Theorem rmo6

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rmo5 3383	. . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑))
2		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐴
3		rmo6.1	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜑
4	2, 3	nfrexw 3310	. . . . . . . 8 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 𝜑
5	4	nfn 1889	. . . . . . 7 ⊢ Ⅎ𝑦 ¬ ∃𝑥 ∈ 𝐴 𝜑
6		idd 25	. . . . . . . 8 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐴))
7		df-rex 3087	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
8	7	notbii 323	. . . . . . . . . 10 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
9		alnex 1813	. . . . . . . . . 10 ⊢ (∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
10	8, 9	bitr4i 281	. . . . . . . . 9 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
11		pm2.21 124	. . . . . . . . . . . 12 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝜑) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦))
12	11	expd 421	. . . . . . . . . . 11 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
13	12	alimi 1843	. . . . . . . . . 10 ⊢ (∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑) → ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
14	13	ralrid 3084	. . . . . . . . 9 ⊢ (∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑) → ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))
15	10, 14	sylbi 220	. . . . . . . 8 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))
16	6, 15	jctird 536	. . . . . . 7 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → (𝑦 ∈ 𝐴 → (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))))
17	5, 16	eximd 2252	. . . . . 6 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → (∃𝑦 𝑦 ∈ 𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))))
18		df-rex 3087	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)))
19	17, 18	imbitrrdi 255	. . . . 5 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → (∃𝑦 𝑦 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)))
20	3	reu6f 35755	. . . . . . 7 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦))
21		biimp 218	. . . . . . . . 9 ⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦))
22	21	ralimi 3099	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))
23	22	reximi 3100	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))
24	20, 23	sylbi 220	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))
25	24	a1d 26	. . . . 5 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (∃𝑦 𝑦 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)))
26	19, 25	ja 188	. . . 4 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑) → (∃𝑦 𝑦 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)))
27	1, 26	sylbi 220	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 → (∃𝑦 𝑦 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)))
28		eleq1w 2843	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
29	28	cbvexvw 2069	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ 𝐴)
30		exsimpl 1900	. . . . . . . 8 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥 𝑥 ∈ 𝐴)
31	30	con3i 155	. . . . . . 7 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 → ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
32	29, 31	sylnbir 334	. . . . . 6 ⊢ (¬ ∃𝑦 𝑦 ∈ 𝐴 → ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
33	32, 8	sylibr 237	. . . . 5 ⊢ (¬ ∃𝑦 𝑦 ∈ 𝐴 → ¬ ∃𝑥 ∈ 𝐴 𝜑)
34		nrexrmo 3384	. . . . 5 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)
35	33, 34	syl 18	. . . 4 ⊢ (¬ ∃𝑦 𝑦 ∈ 𝐴 → ∃*𝑥 ∈ 𝐴 𝜑)
36	3	rmo2i 3834	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) → ∃*𝑥 ∈ 𝐴 𝜑)
37	35, 36	ja 188	. . 3 ⊢ ((∃𝑦 𝑦 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)) → ∃*𝑥 ∈ 𝐴 𝜑)
38	27, 37	impbii 212	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑦 𝑦 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)))
39		eleq1w 2843	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
40	39	cbvexvw 2069	. . . 4 ⊢ (∃𝑦 𝑦 ∈ 𝐴 ↔ ∃𝑤 𝑤 ∈ 𝐴)
41		eleq1w 2843	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
42	41	cbvexvw 2069	. . . 4 ⊢ (∃𝑤 𝑤 ∈ 𝐴 ↔ ∃𝑧 𝑧 ∈ 𝐴)
43	40, 42	bitri 278	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 ↔ ∃𝑧 𝑧 ∈ 𝐴)
44	43	imbi1i 352	. 2 ⊢ ((∃𝑦 𝑦 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)) ↔ (∃𝑧 𝑧 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)))
45	38, 44	bitri 278	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑧 𝑧 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567  ∃wex 1811  Ⅎwnf 1815   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  ∃!wreu 3363  ∃*wrmo 3364

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 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-10 2177  ax-11 2193  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366

This theorem is used by:  srepdflem2  36005