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Theorem riotasv2d 34561
Description: Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 4904). Special case of riota2f 6672. (Contributed by NM, 2-Mar-2013.)
Hypotheses
Ref Expression
riotasv2d.1 𝑦𝜑
riotasv2d.2 (𝜑𝑦𝐹)
riotasv2d.3 (𝜑 → Ⅎ𝑦𝜒)
riotasv2d.4 (𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))
riotasv2d.5 ((𝜑𝑦 = 𝐸) → (𝜓𝜒))
riotasv2d.6 ((𝜑𝑦 = 𝐸) → 𝐶 = 𝐹)
riotasv2d.7 (𝜑𝐷𝐴)
riotasv2d.8 (𝜑𝐸𝐵)
riotasv2d.9 (𝜑𝜒)
Assertion
Ref Expression
riotasv2d ((𝜑𝐴𝑉) → 𝐷 = 𝐹)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶   𝑦,𝐸   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑥,𝑦)   𝐶(𝑦)   𝐷(𝑥,𝑦)   𝐸(𝑥)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem riotasv2d
StepHypRef Expression
1 elex 3243 . 2 (𝐴𝑉𝐴 ∈ V)
2 riotasv2d.8 . . . 4 (𝜑𝐸𝐵)
32adantr 480 . . 3 ((𝜑𝐴 ∈ V) → 𝐸𝐵)
4 riotasv2d.9 . . . 4 (𝜑𝜒)
54adantr 480 . . 3 ((𝜑𝐴 ∈ V) → 𝜒)
6 eleq1 2718 . . . . . . . 8 (𝑦 = 𝐸 → (𝑦𝐵𝐸𝐵))
76adantl 481 . . . . . . 7 ((𝜑𝑦 = 𝐸) → (𝑦𝐵𝐸𝐵))
8 riotasv2d.5 . . . . . . 7 ((𝜑𝑦 = 𝐸) → (𝜓𝜒))
97, 8anbi12d 747 . . . . . 6 ((𝜑𝑦 = 𝐸) → ((𝑦𝐵𝜓) ↔ (𝐸𝐵𝜒)))
10 riotasv2d.6 . . . . . . 7 ((𝜑𝑦 = 𝐸) → 𝐶 = 𝐹)
1110eqeq2d 2661 . . . . . 6 ((𝜑𝑦 = 𝐸) → (𝐷 = 𝐶𝐷 = 𝐹))
129, 11imbi12d 333 . . . . 5 ((𝜑𝑦 = 𝐸) → (((𝑦𝐵𝜓) → 𝐷 = 𝐶) ↔ ((𝐸𝐵𝜒) → 𝐷 = 𝐹)))
1312adantlr 751 . . . 4 (((𝜑𝐴 ∈ V) ∧ 𝑦 = 𝐸) → (((𝑦𝐵𝜓) → 𝐷 = 𝐶) ↔ ((𝐸𝐵𝜒) → 𝐷 = 𝐹)))
14 riotasv2d.4 . . . . 5 (𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))
15 riotasv2d.7 . . . . 5 (𝜑𝐷𝐴)
1614, 15riotasvd 34560 . . . 4 ((𝜑𝐴 ∈ V) → ((𝑦𝐵𝜓) → 𝐷 = 𝐶))
17 riotasv2d.1 . . . . 5 𝑦𝜑
18 nfv 1883 . . . . 5 𝑦 𝐴 ∈ V
1917, 18nfan 1868 . . . 4 𝑦(𝜑𝐴 ∈ V)
20 nfcvd 2794 . . . 4 ((𝜑𝐴 ∈ V) → 𝑦𝐸)
21 nfvd 1884 . . . . . . 7 (𝜑 → Ⅎ𝑦 𝐸𝐵)
22 riotasv2d.3 . . . . . . 7 (𝜑 → Ⅎ𝑦𝜒)
2321, 22nfand 1866 . . . . . 6 (𝜑 → Ⅎ𝑦(𝐸𝐵𝜒))
24 nfra1 2970 . . . . . . . . 9 𝑦𝑦𝐵 (𝜓𝑥 = 𝐶)
25 nfcv 2793 . . . . . . . . 9 𝑦𝐴
2624, 25nfriota 6660 . . . . . . . 8 𝑦(𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))
2717, 14nfceqdf 2789 . . . . . . . 8 (𝜑 → (𝑦𝐷𝑦(𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))))
2826, 27mpbiri 248 . . . . . . 7 (𝜑𝑦𝐷)
29 riotasv2d.2 . . . . . . 7 (𝜑𝑦𝐹)
3028, 29nfeqd 2801 . . . . . 6 (𝜑 → Ⅎ𝑦 𝐷 = 𝐹)
3123, 30nfimd 1863 . . . . 5 (𝜑 → Ⅎ𝑦((𝐸𝐵𝜒) → 𝐷 = 𝐹))
3231adantr 480 . . . 4 ((𝜑𝐴 ∈ V) → Ⅎ𝑦((𝐸𝐵𝜒) → 𝐷 = 𝐹))
333, 13, 16, 19, 20, 32vtocldf 3287 . . 3 ((𝜑𝐴 ∈ V) → ((𝐸𝐵𝜒) → 𝐷 = 𝐹))
343, 5, 33mp2and 715 . 2 ((𝜑𝐴 ∈ V) → 𝐷 = 𝐹)
351, 34sylan2 490 1 ((𝜑𝐴𝑉) → 𝐷 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wnf 1748  wcel 2030  wnfc 2780  wral 2941  Vcvv 3231  crio 6650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-riotaBAD 34557
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-riota 6651  df-undef 7444
This theorem is referenced by:  riotasv2s  34562  cdleme42b  36083
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