Theorem riotasv3d 39220

Description: A property 𝜒 holding for a representative of a single-valued class expression 𝐶(𝑦) (see e.g. reusv2 5348) 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 3461	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		riotasv3d.7	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 𝜓)
3	2	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ∃𝑦 ∈ 𝐵 𝜓)
4		riotasv3d.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
5		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑦 𝐴 ∈ V
6		riotasv3d.5	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝜒))
7	6	imp 406	. . . . . . . . 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 39216	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶))
12	11	impr 454	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝐷 = 𝐶)
13	12	eqcomd 2742	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝐶 = 𝐷)
14		riotasv3d.4	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝜒 ↔ 𝜃))
15	13, 14	syldan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → (𝜒 ↔ 𝜃))
16	8, 15	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝜃)
17	16	exp45 438	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ V → (𝑦 ∈ 𝐵 → (𝜓 → 𝜃))))
18	4, 5, 17	ralrimd 3241	. . . . 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 239	. . . 4 ⊢ (𝜑 → (𝐴 ∈ V → (∃𝑦 ∈ 𝐵 𝜓 → 𝜃)))
23	22	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → (∃𝑦 ∈ 𝐵 𝜓 → 𝜃))
24	3, 23	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝜃)
25	1, 24	sylan2 593	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  Ⅎwnf 1784   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  Vcvv 3440  ℩crio 7314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-riotaBAD 39213

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-riota 7315  df-undef 8215

This theorem is referenced by:  cdlemefs32sn1aw  40674  cdleme43fsv1snlem  40680  cdleme41sn3a  40693  cdleme40m  40727  cdleme40n  40728  cdlemkid  41196  dihvalcqpre  41495