Theorem riotasv3d 39459

Description: A property 𝜒 holding for a representative of a single-valued class expression 𝐶(𝑦) (see e.g. reusv2 5339) 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 3453	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		riotasv3d.7	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 𝜓)
3	2	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ∃𝑦 ∈ 𝐵 𝜓)
4		riotasv3d.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
5		nfv 1921	. . . . . 6 ⊢ Ⅎ𝑦 𝐴 ∈ V
6		riotasv3d.5	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝜒))
7	6	imp 407	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝜓)) → 𝜒)
8	7	adantrl 722	. . . . . . . 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 39455	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶))
12	11	impr 455	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝐷 = 𝐶)
13	12	eqcomd 2746	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝐶 = 𝐷)
14		riotasv3d.4	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝜒 ↔ 𝜃))
15	13, 14	syldan 597	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → (𝜒 ↔ 𝜃))
16	8, 15	mpbid 233	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝜃)
17	16	exp45 439	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ V → (𝑦 ∈ 𝐵 → (𝜓 → 𝜃))))
18	4, 5, 17	ralrimd 3245	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ V → ∀𝑦 ∈ 𝐵 (𝜓 → 𝜃)))
19		riotasv3d.2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑦𝜃)
20		r19.23t 3236	. . . . . 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 240	. . . 4 ⊢ (𝜑 → (𝐴 ∈ V → (∃𝑦 ∈ 𝐵 𝜓 → 𝜃)))
23	22	imp 407	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → (∃𝑦 ∈ 𝐵 𝜓 → 𝜃))
24	3, 23	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝜃)
25	1, 24	sylan2 599	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  Ⅎwnf 1790   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064  Vcvv 3432  ℩crio 7319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-riotaBAD 39452

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-riota 7320  df-undef 8220

This theorem is referenced by:  cdlemefs32sn1aw  40913  cdleme43fsv1snlem  40919  cdleme41sn3a  40932  cdleme40m  40966  cdleme40n  40967  cdlemkid  41435  dihvalcqpre  41734