Theorem riotasv3d 36711

Description: A property 𝜒 holding for a representative of a single-valued class expression 𝐶(𝑦) (see e.g. reusv2 5296) also holds for its description binder 𝐷 (in the form of property 𝜃). (Contributed by NM, 5-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
riotasv3d.1	⊢ Ⅎ𝑦𝜑
riotasv3d.2	⊢ (𝜑 → Ⅎ𝑦𝜃)
riotasv3d.3	⊢ (𝜑 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)))
riotasv3d.4	⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝜒 ↔ 𝜃))
riotasv3d.5	⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝜒))
riotasv3d.6	⊢ (𝜑 → 𝐷 ∈ 𝐴)
riotasv3d.7	⊢ (𝜑 → ∃𝑦 ∈ 𝐵 𝜓)

Assertion

Ref	Expression
riotasv3d	⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝜃)

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

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

Proof of Theorem riotasv3d

Step	Hyp	Ref	Expression
1		elex 3426	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		riotasv3d.7	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 𝜓)
3	2	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ∃𝑦 ∈ 𝐵 𝜓)
4		riotasv3d.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
5		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑦 𝐴 ∈ V
6		riotasv3d.5	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝜒))
7	6	imp 410	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝜓)) → 𝜒)
8	7	adantrl 716	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝜒)
9		riotasv3d.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)))
10		riotasv3d.6	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ∈ 𝐴)
11	9, 10	riotasvd 36707	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶))
12	11	impr 458	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝐷 = 𝐶)
13	12	eqcomd 2743	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝐶 = 𝐷)
14		riotasv3d.4	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝜒 ↔ 𝜃))
15	13, 14	syldan 594	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → (𝜒 ↔ 𝜃))
16	8, 15	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝜃)
17	16	exp45 442	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ V → (𝑦 ∈ 𝐵 → (𝜓 → 𝜃))))
18	4, 5, 17	ralrimd 3139	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ V → ∀𝑦 ∈ 𝐵 (𝜓 → 𝜃)))
19		riotasv3d.2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑦𝜃)
20		r19.23t 3232	. . . . . 6 ⊢ (Ⅎ𝑦𝜃 → (∀𝑦 ∈ 𝐵 (𝜓 → 𝜃) ↔ (∃𝑦 ∈ 𝐵 𝜓 → 𝜃)))
21	19, 20	syl 17	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝜓 → 𝜃) ↔ (∃𝑦 ∈ 𝐵 𝜓 → 𝜃)))
22	18, 21	sylibd 242	. . . 4 ⊢ (𝜑 → (𝐴 ∈ V → (∃𝑦 ∈ 𝐵 𝜓 → 𝜃)))
23	22	imp 410	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → (∃𝑦 ∈ 𝐵 𝜓 → 𝜃))
24	3, 23	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝜃)
25	1, 24	sylan2 596	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543  Ⅎwnf 1791   ∈ wcel 2110  ∀wral 3061  ∃wrex 3062  Vcvv 3408  ℩crio 7169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-riotaBAD 36704

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-riota 7170  df-undef 8015

This theorem is referenced by:  cdlemefs32sn1aw  38165  cdleme43fsv1snlem  38171  cdleme41sn3a  38184  cdleme40m  38218  cdleme40n  38219  cdlemkid  38687  dihvalcqpre  38986