Theorem riotasvd 35574

Description: Deduction version of riotasv 35577. (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 473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)))
3		riotasvd.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ 𝐴)
4	3	adantr 473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐷 ∈ 𝐴)
5	2, 4	eqeltrrd 2860	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) ∈ 𝐴)
6		riotaclbgBAD 35572	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) ∈ 𝐴))
7	6	adantl 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) ∈ 𝐴))
8	5, 7	mpbird 249	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
9		riotasbc 6950	. . . . . 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 2775	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐶 ↔ 𝑧 = 𝐶))
12	11	imbi2d 333	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝜓 → 𝑥 = 𝐶) ↔ (𝜓 → 𝑧 = 𝐶)))
13	12	ralbidv 3140	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 (𝜓 → 𝑧 = 𝐶)))
14		nfra1 3162	. . . . . . . . . 10 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)
15		nfcv 2925	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐴
16	14, 15	nfriota 6944	. . . . . . . . 9 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
17	16	nfeq2 2940	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑧 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
18		eqeq1 2775	. . . . . . . . 9 ⊢ (𝑧 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) → (𝑧 = 𝐶 ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶))
19	18	imbi2d 333	. . . . . . . 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 3171	. . . . . . 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 3708	. . . . . 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 224	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ∀𝑦 ∈ 𝐵 (𝜓 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶))
24		rsp 3148	. . . 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 402	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶))
27	2	eqeq1d 2773	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝐷 = 𝐶 ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶))
28	26, 27	sylibrd 251	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1508   ∈ wcel 2051  ∀wral 3081  ∃!wreu 3083  [wsbc 3674  ℩crio 6934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-riotaBAD 35571

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-iota 6149  df-fun 6187  df-fv 6193  df-riota 6935  df-undef 7740

This theorem is referenced by:  riotasv2d  35575  riotasv  35577  riotasv3d  35578  cdleme32a  37059