Theorem riotasvd 34242

Description: Deduction version of riotasv 34245. (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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)))
3		riotasvd.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ 𝐴)
4	3	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐷 ∈ 𝐴)
5	2, 4	eqeltrrd 2702	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) ∈ 𝐴)
6		riotaclbgBAD 34240	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) ∈ 𝐴))
7	6	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) ∈ 𝐴))
8	5, 7	mpbird 247	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
9		riotasbc 6626	. . . . . 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 2626	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐶 ↔ 𝑧 = 𝐶))
12	11	imbi2d 330	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝜓 → 𝑥 = 𝐶) ↔ (𝜓 → 𝑧 = 𝐶)))
13	12	ralbidv 2986	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 (𝜓 → 𝑧 = 𝐶)))
14		nfra1 2941	. . . . . . . . . 10 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)
15		nfcv 2764	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐴
16	14, 15	nfriota 6620	. . . . . . . . 9 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
17	16	nfeq2 2780	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑧 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
18		eqeq1 2626	. . . . . . . . 9 ⊢ (𝑧 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) → (𝑧 = 𝐶 ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶))
19	18	imbi2d 330	. . . . . . . 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 2983	. . . . . . 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 3469	. . . . . 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 222	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ∀𝑦 ∈ 𝐵 (𝜓 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶))
24		rsp 2929	. . . 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 447	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶))
27	2	eqeq1d 2624	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝐷 = 𝐶 ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶))
28	26, 27	sylibrd 249	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃!wreu 2914  [wsbc 3435  ℩crio 6610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-undef 7399

This theorem is referenced by:  riotasv2d  34243  riotasv  34245  riotasv3d  34246  cdleme32a  35729