Theorem rmo3 3056

Description: Restricted at-most-one quantifier using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)

Hypothesis

Ref	Expression
rmo2.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
rmo3	⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem rmo3

Step	Hyp	Ref	Expression
1		df-rmo 2463	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		sban 1955	. . . . . . . . . . 11 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))
3		clelsb1 2282	. . . . . . . . . . . 12 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
4	3	anbi1i 458	. . . . . . . . . . 11 ⊢ (([𝑦 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))
5	2, 4	bitri 184	. . . . . . . . . 10 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))
6	5	anbi2i 457	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)))
7		an4 586	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
8		ancom 266	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
9	8	anbi1i 458	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
10	6, 7, 9	3bitri 206	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
11	10	imbi1i 238	. . . . . . 7 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝑦) ↔ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑥 = 𝑦))
12		impexp 263	. . . . . . 7 ⊢ ((((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑥 = 𝑦) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
13		impexp 263	. . . . . . 7 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
14	11, 12, 13	3bitri 206	. . . . . 6 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝑦) ↔ (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
15	14	albii 1470	. . . . 5 ⊢ (∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝑦) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
16		df-ral 2460	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
17		r19.21v 2554	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
18	15, 16, 17	3bitr2i 208	. . . 4 ⊢ (∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
19	18	albii 1470	. . 3 ⊢ (∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
20		nfv 1528	. . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
21		rmo2.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
22	20, 21	nfan 1565	. . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)
23	22	mo3 2080	. . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝑦))
24		df-ral 2460	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
25	19, 23, 24	3bitr4i 212	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
26	1, 25	bitri 184	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1351  Ⅎwnf 1460  [wsb 1762  ∃*wmo 2027   ∈ wcel 2148  ∀wral 2455  ∃*wrmo 2458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-cleq 2170  df-clel 2173  df-ral 2460  df-rmo 2463

This theorem is referenced by:  disjiun  4000