Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  riotasv3d Structured version   Visualization version   GIF version

Theorem riotasv3d 36296
 Description: A property 𝜒 holding for a representative of a single-valued class expression 𝐶(𝑦) (see e.g. reusv2 5270) 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 𝑦𝜑
riotasv3d.2 (𝜑 → Ⅎ𝑦𝜃)
riotasv3d.3 (𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))
riotasv3d.4 ((𝜑𝐶 = 𝐷) → (𝜒𝜃))
riotasv3d.5 (𝜑 → ((𝑦𝐵𝜓) → 𝜒))
riotasv3d.6 (𝜑𝐷𝐴)
riotasv3d.7 (𝜑 → ∃𝑦𝐵 𝜓)
Assertion
Ref Expression
riotasv3d ((𝜑𝐴𝑉) → 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
StepHypRef Expression
1 elex 3459 . 2 (𝐴𝑉𝐴 ∈ V)
2 riotasv3d.7 . . . 4 (𝜑 → ∃𝑦𝐵 𝜓)
32adantr 484 . . 3 ((𝜑𝐴 ∈ V) → ∃𝑦𝐵 𝜓)
4 riotasv3d.1 . . . . . 6 𝑦𝜑
5 nfv 1915 . . . . . 6 𝑦 𝐴 ∈ V
6 riotasv3d.5 . . . . . . . . . 10 (𝜑 → ((𝑦𝐵𝜓) → 𝜒))
76imp 410 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝜓)) → 𝜒)
87adantrl 715 . . . . . . . 8 ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦𝐵𝜓))) → 𝜒)
9 riotasv3d.3 . . . . . . . . . . . 12 (𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))
10 riotasv3d.6 . . . . . . . . . . . 12 (𝜑𝐷𝐴)
119, 10riotasvd 36292 . . . . . . . . . . 11 ((𝜑𝐴 ∈ V) → ((𝑦𝐵𝜓) → 𝐷 = 𝐶))
1211impr 458 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦𝐵𝜓))) → 𝐷 = 𝐶)
1312eqcomd 2804 . . . . . . . . 9 ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦𝐵𝜓))) → 𝐶 = 𝐷)
14 riotasv3d.4 . . . . . . . . 9 ((𝜑𝐶 = 𝐷) → (𝜒𝜃))
1513, 14syldan 594 . . . . . . . 8 ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦𝐵𝜓))) → (𝜒𝜃))
168, 15mpbid 235 . . . . . . 7 ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦𝐵𝜓))) → 𝜃)
1716exp45 442 . . . . . 6 (𝜑 → (𝐴 ∈ V → (𝑦𝐵 → (𝜓𝜃))))
184, 5, 17ralrimd 3182 . . . . 5 (𝜑 → (𝐴 ∈ V → ∀𝑦𝐵 (𝜓𝜃)))
19 riotasv3d.2 . . . . . 6 (𝜑 → Ⅎ𝑦𝜃)
20 r19.23t 3272 . . . . . 6 (Ⅎ𝑦𝜃 → (∀𝑦𝐵 (𝜓𝜃) ↔ (∃𝑦𝐵 𝜓𝜃)))
2119, 20syl 17 . . . . 5 (𝜑 → (∀𝑦𝐵 (𝜓𝜃) ↔ (∃𝑦𝐵 𝜓𝜃)))
2218, 21sylibd 242 . . . 4 (𝜑 → (𝐴 ∈ V → (∃𝑦𝐵 𝜓𝜃)))
2322imp 410 . . 3 ((𝜑𝐴 ∈ V) → (∃𝑦𝐵 𝜓𝜃))
243, 23mpd 15 . 2 ((𝜑𝐴 ∈ V) → 𝜃)
251, 24sylan2 595 1 ((𝜑𝐴𝑉) → 𝜃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441  ℩crio 7097 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448  ax-riotaBAD 36289 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6286  df-fun 6329  df-fv 6335  df-riota 7098  df-undef 7929 This theorem is referenced by:  cdlemefs32sn1aw  37750  cdleme43fsv1snlem  37756  cdleme41sn3a  37769  cdleme40m  37803  cdleme40n  37804  cdlemkid  38272  dihvalcqpre  38571
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