Theorem rmo4 3660

Description: Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.)

Hypothesis

Ref	Expression
rmo4.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rmo4	⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))

Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem rmo4

Step	Hyp	Ref	Expression
1		df-rmo 3071	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		an4 652	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝜑 ∧ 𝜓)))
3		ancom 460	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
4	3	anbi1i 623	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝜑 ∧ 𝜓)) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜑 ∧ 𝜓)))
5	2, 4	bitri 274	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜑 ∧ 𝜓)))
6	5	imbi1i 349	. . . . . . 7 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)) → 𝑥 = 𝑦) ↔ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜑 ∧ 𝜓)) → 𝑥 = 𝑦))
7		impexp 450	. . . . . . 7 ⊢ ((((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜑 ∧ 𝜓)) → 𝑥 = 𝑦) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
8		impexp 450	. . . . . . 7 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) ↔ (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))))
9	6, 7, 8	3bitri 296	. . . . . 6 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)) → 𝑥 = 𝑦) ↔ (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))))
10	9	albii 1823	. . . . 5 ⊢ (∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)) → 𝑥 = 𝑦) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))))
11		df-ral 3068	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝐴 → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))))
12		r19.21v 3100	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝐴 → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
13	10, 11, 12	3bitr2i 298	. . . 4 ⊢ (∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
14	13	albii 1823	. . 3 ⊢ (∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
15		eleq1w 2821	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
16		rmo4.1	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
17	15, 16	anbi12d 630	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓)))
18	17	mo4 2566	. . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)) → 𝑥 = 𝑦))
19		df-ral 3068	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
20	14, 18, 19	3bitr4i 302	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
21	1, 20	bitri 274	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   ∈ wcel 2108  ∃*wmo 2538  ∀wral 3063  ∃*wrmo 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-mo 2540  df-clel 2817  df-ral 3068  df-rmo 3071

This theorem is referenced by:  reu4  3661  disjor  5050  somo  5531  supmo  9141  infmo  9184  sqrmo  14891  catideu  17301  poslubmo  18044  posglbmo  18045  mgmidmo  18259  mndinvmod  18330  lspextmo  20233  evlseu  21203  ply1divmo  25205  2sqmo  26490  tghilberti2  26903  foot  26987  mideu  27003  cvmliftmo  33146  nnasmo  33596  hilbert1.2  34384  poimirlem1  35705  poimirlem13  35717  poimirlem14  35718  poimirlem18  35722  poimirlem21  35725  inecmo  36414  idomsubgmo  40939