Theorem rmo4 2999

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 2518	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		an4 588	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝜑 ∧ 𝜓)))
3		ancom 266	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
4	3	anbi1i 458	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝜑 ∧ 𝜓)) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜑 ∧ 𝜓)))
5	2, 4	bitri 184	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜑 ∧ 𝜓)))
6	5	imbi1i 238	. . . . . . 7 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)) → 𝑥 = 𝑦) ↔ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜑 ∧ 𝜓)) → 𝑥 = 𝑦))
7		impexp 263	. . . . . . 7 ⊢ ((((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜑 ∧ 𝜓)) → 𝑥 = 𝑦) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
8		impexp 263	. . . . . . 7 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) ↔ (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))))
9	6, 7, 8	3bitri 206	. . . . . 6 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)) → 𝑥 = 𝑦) ↔ (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))))
10	9	albii 1518	. . . . 5 ⊢ (∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)) → 𝑥 = 𝑦) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))))
11		df-ral 2515	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝐴 → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) ↔ ∀𝑦(𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))))
12		r19.21v 2609	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝐴 → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
13	10, 11, 12	3bitr2i 208	. . . 4 ⊢ (∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
14	13	albii 1518	. . 3 ⊢ (∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
15		eleq1 2294	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
16		rmo4.1	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
17	15, 16	anbi12d 473	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓)))
18	17	mo4 2141	. . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥∀𝑦(((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)) → 𝑥 = 𝑦))
19		df-ral 2515	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
20	14, 18, 19	3bitr4i 212	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
21	1, 20	bitri 184	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1395  ∃*wmo 2080   ∈ wcel 2202  ∀wral 2510  ∃*wrmo 2513

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-cleq 2224  df-clel 2227  df-ral 2515  df-rmo 2518

This theorem is referenced by:  reu4  3000  disjnim  4078  supmoti  7191  lteupri  7836  elrealeu  8048  rereceu  8108  exbtwnz  10509  rsqrmo  11587  divalglemeunn  12481  divalglemeuneg  12483  bezoutlemeu  12577  pw2dvdseu  12739  mgmidmo  13454  mndinvmod  13527  dedekindeu  15346  dedekindicclemicc  15355