Theorem riotasvd 38339

Description: Deduction version of riotasv 38342. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
riotasvd.1	⊢ (𝜑 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)))
riotasvd.2	⊢ (𝜑 → 𝐷 ∈ 𝐴)

Assertion

Ref	Expression
riotasvd	⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵   𝑥,𝐶   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝐵(𝑦)   𝐶(𝑦)   𝐷(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem riotasvd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		riotasvd.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)))
2	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)))
3		riotasvd.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ 𝐴)
4	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐷 ∈ 𝐴)
5	2, 4	eqeltrrd 2828	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) ∈ 𝐴)
6		riotaclbgBAD 38337	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) ∈ 𝐴))
7	6	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) ∈ 𝐴))
8	5, 7	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
9		riotasbc 7380	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) → [(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) / 𝑥]∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
10	8, 9	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) / 𝑥]∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
11		eqeq1 2730	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐶 ↔ 𝑧 = 𝐶))
12	11	imbi2d 340	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝜓 → 𝑥 = 𝐶) ↔ (𝜓 → 𝑧 = 𝐶)))
13	12	ralbidv 3171	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 (𝜓 → 𝑧 = 𝐶)))
14		nfra1 3275	. . . . . . . . . 10 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)
15		nfcv 2897	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐴
16	14, 15	nfriota 7374	. . . . . . . . 9 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
17	16	nfeq2 2914	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑧 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
18		eqeq1 2730	. . . . . . . . 9 ⊢ (𝑧 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) → (𝑧 = 𝐶 ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶))
19	18	imbi2d 340	. . . . . . . 8 ⊢ (𝑧 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) → ((𝜓 → 𝑧 = 𝐶) ↔ (𝜓 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶)))
20	17, 19	ralbid 3264	. . . . . . 7 ⊢ (𝑧 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) → (∀𝑦 ∈ 𝐵 (𝜓 → 𝑧 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 (𝜓 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶)))
21	13, 20	sbcie2g 3814	. . . . . 6 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) ∈ 𝐴 → ([(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) / 𝑥]∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 (𝜓 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶)))
22	5, 21	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ([(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) / 𝑥]∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 (𝜓 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶)))
23	10, 22	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ∀𝑦 ∈ 𝐵 (𝜓 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶))
24		rsp 3238	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 (𝜓 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶) → (𝑦 ∈ 𝐵 → (𝜓 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶)))
25	23, 24	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ 𝐵 → (𝜓 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶)))
26	25	impd 410	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶))
27	2	eqeq1d 2728	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝐷 = 𝐶 ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶))
28	26, 27	sylibrd 259	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3055  ∃!wreu 3368  [wsbc 3772  ℩crio 7360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-riotaBAD 38336

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-riota 7361  df-undef 8259

This theorem is referenced by:  riotasv2d  38340  riotasv  38342  riotasv3d  38343  cdleme32a  39825