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Theorem riotasv3d 33067
Description: A property 𝜒 holding for a representative of a single-valued class expression 𝐶(𝑦) (see e.g. reusv2 4795) 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 3184 . 2 (𝐴𝑉𝐴 ∈ V)
2 riotasv3d.7 . . . 4 (𝜑 → ∃𝑦𝐵 𝜓)
32adantr 479 . . 3 ((𝜑𝐴 ∈ V) → ∃𝑦𝐵 𝜓)
4 riotasv3d.1 . . . . . 6 𝑦𝜑
5 nfv 1829 . . . . . 6 𝑦 𝐴 ∈ V
6 riotasv3d.5 . . . . . . . . . 10 (𝜑 → ((𝑦𝐵𝜓) → 𝜒))
76imp 443 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝜓)) → 𝜒)
87adantrl 747 . . . . . . . 8 ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦𝐵𝜓))) → 𝜒)
9 riotasv3d.3 . . . . . . . . . . . 12 (𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))
10 riotasv3d.6 . . . . . . . . . . . 12 (𝜑𝐷𝐴)
119, 10riotasvd 33063 . . . . . . . . . . 11 ((𝜑𝐴 ∈ V) → ((𝑦𝐵𝜓) → 𝐷 = 𝐶))
1211impr 646 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦𝐵𝜓))) → 𝐷 = 𝐶)
1312eqcomd 2615 . . . . . . . . 9 ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦𝐵𝜓))) → 𝐶 = 𝐷)
14 riotasv3d.4 . . . . . . . . 9 ((𝜑𝐶 = 𝐷) → (𝜒𝜃))
1513, 14syldan 485 . . . . . . . 8 ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦𝐵𝜓))) → (𝜒𝜃))
168, 15mpbid 220 . . . . . . 7 ((𝜑 ∧ (𝐴 ∈ V ∧ (𝑦𝐵𝜓))) → 𝜃)
1716exp45 639 . . . . . 6 (𝜑 → (𝐴 ∈ V → (𝑦𝐵 → (𝜓𝜃))))
184, 5, 17ralrimd 2941 . . . . 5 (𝜑 → (𝐴 ∈ V → ∀𝑦𝐵 (𝜓𝜃)))
19 riotasv3d.2 . . . . . 6 (𝜑 → Ⅎ𝑦𝜃)
20 r19.23t 3002 . . . . . 6 (Ⅎ𝑦𝜃 → (∀𝑦𝐵 (𝜓𝜃) ↔ (∃𝑦𝐵 𝜓𝜃)))
2119, 20syl 17 . . . . 5 (𝜑 → (∀𝑦𝐵 (𝜓𝜃) ↔ (∃𝑦𝐵 𝜓𝜃)))
2218, 21sylibd 227 . . . 4 (𝜑 → (𝐴 ∈ V → (∃𝑦𝐵 𝜓𝜃)))
2322imp 443 . . 3 ((𝜑𝐴 ∈ V) → (∃𝑦𝐵 𝜓𝜃))
243, 23mpd 15 . 2 ((𝜑𝐴 ∈ V) → 𝜃)
251, 24sylan2 489 1 ((𝜑𝐴𝑉) → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wnf 1698  wcel 1976  wral 2895  wrex 2896  Vcvv 3172  crio 6488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-riotaBAD 33060
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-riota 6489  df-undef 7263
This theorem is referenced by:  cdlemefs32sn1aw  34523  cdleme43fsv1snlem  34529  cdleme41sn3a  34542  cdleme40m  34576  cdleme40n  34577  cdlemkid  35045  dihvalcqpre  35345
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