Theorem riota5f 7394

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 3147	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝐵))
3		riota5f.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
4		trud 1552	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → ⊤)
5		reu6i 3725	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦)) → ∃!𝑥 ∈ 𝐴 𝜓)
6	5	adantl 483	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → ∃!𝑥 ∈ 𝐴 𝜓)
7		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝜑
8		nfv 1918	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
9		nfra1 3282	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦)
10	8, 9	nfan 1903	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))
11	7, 10	nfan 1903	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦)))
12		nfcvd 2905	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → Ⅎ𝑥𝑦)
13		nfvd 1919	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → Ⅎ𝑥⊤)
14		simprl 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → 𝑦 ∈ 𝐴)
15		simpr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
16		simplrr 777	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))
17		simplrl 776	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑦 ∈ 𝐴)
18	15, 17	eqeltrd 2834	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝑥 ∈ 𝐴)
19		rsp 3245	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝑥 = 𝑦)))
20	16, 18, 19	sylc 65	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝑥 = 𝑦))
21	15, 20	mpbird 257	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → 𝜓)
22		trud 1552	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → ⊤)
23	21, 22	2thd 265	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ 𝑥 = 𝑦) → (𝜓 ↔ ⊤))
24	11, 12, 13, 14, 23	riota2df 7389	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (⊤ ↔ (℩𝑥 ∈ 𝐴 𝜓) = 𝑦))
25	6, 24	mpdan 686	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → (⊤ ↔ (℩𝑥 ∈ 𝐴 𝜓) = 𝑦))
26	4, 25	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦))) → (℩𝑥 ∈ 𝐴 𝜓) = 𝑦)
27	26	expr 458	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥 ∈ 𝐴 𝜓) = 𝑦))
28	27	ralrimiva 3147	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥 ∈ 𝐴 𝜓) = 𝑦))
29		rspsbc 3874	. . . 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 65	. . 3 ⊢ (𝜑 → [𝐵 / 𝑦](∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥 ∈ 𝐴 𝜓) = 𝑦))
31		nfcvd 2905	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝑦)
32		riota5f.1	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝐵)
33	31, 32	nfeqd 2914	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝐵)
34	7, 33	nfan1 2194	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 = 𝐵)
35		simpr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
36	35	eqeq2d 2744	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝑥 = 𝐵))
37	36	bibi2d 343	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → ((𝜓 ↔ 𝑥 = 𝑦) ↔ (𝜓 ↔ 𝑥 = 𝐵)))
38	34, 37	ralbid 3271	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝐵)))
39	35	eqeq2d 2744	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → ((℩𝑥 ∈ 𝐴 𝜓) = 𝑦 ↔ (℩𝑥 ∈ 𝐴 𝜓) = 𝐵))
40	38, 39	imbi12d 345	. . . 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 3823	. . 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 231	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝑥 = 𝐵) → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵))
43	2, 42	mpd 15	1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  ⊤wtru 1543   ∈ wcel 2107  Ⅎwnfc 2884  ∀wral 3062  ∃!wreu 3375  [wsbc 3778  ℩crio 7364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-reu 3378  df-v 3477  df-sbc 3779  df-un 3954  df-in 3956  df-ss 3966  df-sn 4630  df-pr 4632  df-uni 4910  df-iota 6496  df-riota 7365

This theorem is referenced by:  riota5  7395