Theorem rmo3f 2881

Description: Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)

Hypotheses

Ref	Expression
rmo3f.1	⊢ Ⅎ𝑥𝐴
rmo3f.2	⊢ Ⅎ𝑦𝐴
rmo3f.3	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
rmo3f	⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem rmo3f

Step	Hyp	Ref	Expression
1		df-rmo 2424	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		sban 1928	. . . . . . . . . . 11 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))
3		rmo3f.1	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝐴
4	3	clelsb3f 2285	. . . . . . . . . . . 12 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
5	4	anbi1i 453	. . . . . . . . . . 11 ⊢ (([𝑦 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))
6	2, 5	bitri 183	. . . . . . . . . 10 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))
7	6	anbi2i 452	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)))
8		an4 575	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
9		ancom 264	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
10	9	anbi1i 453	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
11	7, 8, 10	3bitri 205	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
12	11	imbi1i 237	. . . . . . 7 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝑦) ↔ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑥 = 𝑦))
13		impexp 261	. . . . . . 7 ⊢ ((((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑥 = 𝑦) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
14		impexp 261	. . . . . . 7 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
15	12, 13, 14	3bitri 205	. . . . . 6 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝑦) ↔ (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
16	15	albii 1446	. . . . 5 ⊢ (∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝑦) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
17		df-ral 2421	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
18		rmo3f.2	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
19	18	nfcri 2275	. . . . . 6 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
20	19	r19.21 2508	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
21	16, 17, 20	3bitr2i 207	. . . 4 ⊢ (∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
22	21	albii 1446	. . 3 ⊢ (∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
23		rmo3f.3	. . . . 5 ⊢ Ⅎ𝑦𝜑
24	19, 23	nfan 1544	. . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)
25	24	mo3 2053	. . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝑦))
26		df-ral 2421	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
27	22, 25, 26	3bitr4i 211	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
28	1, 27	bitri 183	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329  Ⅎwnf 1436   ∈ wcel 1480  [wsb 1735  ∃*wmo 2000  Ⅎwnfc 2268  ∀wral 2416  ∃*wrmo 2419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rmo 2424

This theorem is referenced by:  rmo4f  2882