Theorem riotasv3d 37830

Description: A property 𝜒 holding for a representative of a single-valued class expression 𝐶(𝑦) (see e.g. reusv2 5402) 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 3493	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		riotasv3d.7	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝐵 𝜓)
3	2	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ∃𝑦 ∈ 𝐵 𝜓)
4		riotasv3d.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
5		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑦 𝐴 ∈ V
6		riotasv3d.5	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝜒))
7	6	imp 408	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝜓)) → 𝜒)
8	7	adantrl 715	. . . . . . . 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 37826	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶))
12	11	impr 456	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝐷 = 𝐶)
13	12	eqcomd 2739	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝐶 = 𝐷)
14		riotasv3d.4	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝜒 ↔ 𝜃))
15	13, 14	syldan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → (𝜒 ↔ 𝜃))
16	8, 15	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦 ∈ 𝐵 ∧ 𝜓))) → 𝜃)
17	16	exp45 440	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ V → (𝑦 ∈ 𝐵 → (𝜓 → 𝜃))))
18	4, 5, 17	ralrimd 3262	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ V → ∀𝑦 ∈ 𝐵 (𝜓 → 𝜃)))
19		riotasv3d.2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑦𝜃)
20		r19.23t 3253	. . . . . 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 238	. . . 4 ⊢ (𝜑 → (𝐴 ∈ V → (∃𝑦 ∈ 𝐵 𝜓 → 𝜃)))
23	22	imp 408	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → (∃𝑦 ∈ 𝐵 𝜓 → 𝜃))
24	3, 23	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ V) → 𝜃)
25	1, 24	sylan2 594	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  Ⅎwnf 1786   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  Vcvv 3475  ℩crio 7364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-riotaBAD 37823

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-riota 7365  df-undef 8258

This theorem is referenced by:  cdlemefs32sn1aw  39285  cdleme43fsv1snlem  39291  cdleme41sn3a  39304  cdleme40m  39338  cdleme40n  39339  cdlemkid  39807  dihvalcqpre  40106