Theorem riota5f 5512

Description: A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
riota5f.1	⊢ (𝜑 → Ⅎ𝑥𝐵)
riota5f.2	⊢ (𝜑 → 𝐵 ∈ 𝐴)
riota5f.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝑥 = 𝐵))

Assertion

Ref	Expression
riota5f	⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)

Proof of Theorem riota5f

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		riota5f.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝑥 = 𝐵))
2	1	ralrimiva 2434	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝐵))
3		riota5f.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
4		a1tru 1300	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → ⊤)
5		reu6i 2783	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦)) → ∃!𝑥 ∈ 𝐴 𝜓)
6	5	adantl 271	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → ∃!𝑥 ∈ 𝐴 𝜓)
7		nfv 1461	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝜑
8		nfv 1461	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
9		nfra1 2397	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦)
10	8, 9	nfan 1497	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))
11	7, 10	nfan 1497	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦)))
12		nfcvd 2220	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → Ⅎ𝑥𝑦)
13		nfvd 1462	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → Ⅎ𝑥⊤)
14		simprl 497	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → 𝑦 ∈ 𝐴)
15		simpr 108	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
16		simplrr 502	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))
17		simplrl 501	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑦 ∈ 𝐴)
18	15, 17	eqeltrd 2155	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥 ∈ 𝐴)
19		rsp 2411	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝑥 = 𝑦)))
20	16, 18, 19	sylc 61	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝑥 = 𝑦))
21	15, 20	mpbird 165	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝜓)
22		a1tru 1300	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → ⊤)
23	21, 22	2thd 173	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝜓 ↔ ⊤))
24	11, 12, 13, 14, 23	riota2df 5508	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (⊤ ↔ (℩𝑥 ∈ 𝐴 𝜓) = 𝑦))
25	6, 24	mpdan 412	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → (⊤ ↔ (℩𝑥 ∈ 𝐴 𝜓) = 𝑦))
26	4, 25	mpbid 145	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → (℩𝑥 ∈ 𝐴 𝜓) = 𝑦)
27	26	expr 367	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥 ∈ 𝐴 𝜓) = 𝑦))
28	27	ralrimiva 2434	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥 ∈ 𝐴 𝜓) = 𝑦))
29		rspsbc 2896	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥 ∈ 𝐴 𝜓) = 𝑦) → [𝐵 / 𝑦](∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥 ∈ 𝐴 𝜓) = 𝑦)))
30	3, 28, 29	sylc 61	. . 3 ⊢ (𝜑 → [𝐵 / 𝑦](∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥 ∈ 𝐴 𝜓) = 𝑦))
31		nfcvd 2220	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝑦)
32		riota5f.1	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝐵)
33	31, 32	nfeqd 2233	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝐵)
34	7, 33	nfan1 1496	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 = 𝐵)
35		simpr 108	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
36	35	eqeq2d 2092	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝑥 = 𝐵))
37	36	bibi2d 230	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → ((𝜓 ↔ 𝑥 = 𝑦) ↔ (𝜓 ↔ 𝑥 = 𝐵)))
38	34, 37	ralbid 2366	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝐵)))
39	35	eqeq2d 2092	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → ((℩𝑥 ∈ 𝐴 𝜓) = 𝑦 ↔ (℩𝑥 ∈ 𝐴 𝜓) = 𝐵))
40	38, 39	imbi12d 232	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → ((∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥 ∈ 𝐴 𝜓) = 𝑦) ↔ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝐵) → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)))
41	3, 40	sbcied 2850	. . 3 ⊢ (𝜑 → ([𝐵 / 𝑦](∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥 ∈ 𝐴 𝜓) = 𝑦) ↔ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝐵) → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)))
42	30, 41	mpbid 145	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝐵) → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵))
43	2, 42	mpd 13	1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284  ⊤wtru 1285   ∈ wcel 1433  Ⅎwnfc 2206  ∀wral 2348  ∃!wreu 2350  [wsbc 2815  ℩crio 5487

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-v 2603  df-sbc 2816  df-un 2977  df-sn 3404  df-pr 3405  df-uni 3602  df-iota 4887  df-riota 5488

This theorem is referenced by:  riota5  5513