Theorem rmo3 3845

Description: Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) Avoid ax-13 2407. (Revised by Wolf Lammen, 30-Apr-2023.)

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 3372	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		sban 2117	. . . . . . . . . . 11 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))
3		clelsb1 2893	. . . . . . . . . . 11 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
4	2, 3	bianbi 639	. . . . . . . . . 10 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑))
5	4	anbi2i 635	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)))
6		an4 669	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥]𝜑)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
7		ancom 466	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
8	7	anbi1i 636	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
9	5, 6, 8	3bitri 300	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)))
10	9	imbi1i 352	. . . . . . 7 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝑦) ↔ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑥 = 𝑦))
11		impexp 456	. . . . . . 7 ⊢ ((((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑥 = 𝑦) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
12		impexp 456	. . . . . . 7 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
13	10, 11, 12	3bitri 300	. . . . . 6 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝑦) ↔ (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
14	13	albii 1852	. . . . 5 ⊢ (∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝑦) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
15		df-ral 3083	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
16		r19.21v 3193	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
17	14, 15, 16	3bitr2i 302	. . . 4 ⊢ (∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
18	17	albii 1852	. . 3 ⊢ (∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
19		nfv 1947	. . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
20		rmo2.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
21	19, 20	nfan 1932	. . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)
22	21	mo3 2595	. . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 = 𝑦))
23		df-ral 3083	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
24	18, 22, 23	3bitr4i 306	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
25	1, 24	bitri 278	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  Ⅎwnf 1816  [wsb 2099   ∈ wcel 2146  ∃*wmo 2568  ∀wral 3082  ∃*wrmo 3371

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-10 2179  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-clel 2841  df-ral 3083  df-rmo 3372

This theorem is used by:  poimirlem25  38336  poimirlem26  38337