Theorem riotasv3d 40111

Description: A property 𝜒 holding for a representative of a single-valued class expression 𝐶(𝑦) (see e.g. reusv2 5364) 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 3471	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		riotasv3d.7	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 𝜓)
3	2	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ∃𝑦 ∈ 𝐵 𝜓)
4		riotasv3d.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
5		nfv 1946	. . . . . 6 ⊢ Ⅎ𝑦 𝐴 ∈ V
6		riotasv3d.5	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝜒))
7	6	imp 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝜓)) → 𝜒)
8	7	adantrl 729	. . . . . . . 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 40107	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶))
12	11	impr 460	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝐷 = 𝐶)
13	12	eqcomd 2766	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝐶 = 𝐷)
14		riotasv3d.4	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝜒 ↔ 𝜃))
15	13, 14	syldan 603	. . . . . . . 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 444	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ V → (𝑦 ∈ 𝐵 → (𝜓 → 𝜃))))
18	4, 5, 17	ralrimd 3267	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ V → ∀𝑦 ∈ 𝐵 (𝜓 → 𝜃)))
19		riotasv3d.2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑦𝜃)
20		r19.23t 3258	. . . . . 6 ⊢ (Ⅎ𝑦𝜃 → (∀𝑦 ∈ 𝐵 (𝜓 → 𝜃) ↔ (∃𝑦 ∈ 𝐵 𝜓 → 𝜃)))
21	19, 20	syl 18	. . . . 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 412	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → (∃𝑦 ∈ 𝐵 𝜓 → 𝜃))
24	3, 23	mpd 16	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝜃)
25	1, 24	sylan2 605	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝜃)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  Ⅎwnf 1815   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450  ℩crio 7377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-riotaBAD 40104

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6541  df-fv 6547  df-riota 7378  df-undef 8294

This theorem is used by:  cdlemefs32sn1aw  41565  cdleme43fsv1snlem  41571  cdleme41sn3a  41584  cdleme40m  41618  cdleme40n  41619  cdlemkid  42087  dihvalcqpre  42386