Theorem riotasvd 39619

Description: Deduction version of riotasv 39622. (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 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)))
3		riotasvd.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ 𝐴)
4	3	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐷 ∈ 𝐴)
5	2, 4	eqeltrrd 2870	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) ∈ 𝐴)
6		riotaclbgBAD 39617	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) ∈ 𝐴))
7	6	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) ∈ 𝐴))
8	5, 7	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
9		riotasbc 7386	. . . . . 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 18	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) / 𝑥]∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
11		eqeq1 2773	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐶 ↔ 𝑧 = 𝐶))
12	11	imbi2d 343	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝜓 → 𝑥 = 𝐶) ↔ (𝜓 → 𝑧 = 𝐶)))
13	12	ralbidv 3194	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 (𝜓 → 𝑧 = 𝐶)))
14		nfra1 3295	. . . . . . . . . 10 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)
15		nfcv 2931	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐴
16	14, 15	nfriota 7380	. . . . . . . . 9 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
17	16	nfeq2 2948	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑧 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
18		eqeq1 2773	. . . . . . . . 9 ⊢ (𝑧 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) → (𝑧 = 𝐶 ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶))
19	18	imbi2d 343	. . . . . . . 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 3284	. . . . . . 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 3793	. . . . . 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 18	. . . . 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 235	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ∀𝑦 ∈ 𝐵 (𝜓 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶))
24		rsp 3259	. . . 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 18	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ 𝐵 → (𝜓 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶)))
26	25	impd 415	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶))
27	2	eqeq1d 2771	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝐷 = 𝐶 ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶))
28	26, 27	sylibrd 262	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃!wreu 3374  [wsbc 3753  ℩crio 7367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-riotaBAD 39616

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-riota 7368  df-undef 8268

This theorem is referenced by:  riotasv2d  39620  riotasv  39622  riotasv3d  39623  cdleme32a  41104