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Theorem riotasv2d 33261
 Description: Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 4800). Special case of riota2f 6532. (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 3185 . 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 2676 . . . . . . . 8 (𝑦 = 𝐸 → (𝑦𝐵𝐸𝐵))
76adantl 481 . . . . . . 7 ((𝜑𝑦 = 𝐸) → (𝑦𝐵𝐸𝐵))
8 riotasv2d.5 . . . . . . 7 ((𝜑𝑦 = 𝐸) → (𝜓𝜒))
97, 8anbi12d 743 . . . . . 6 ((𝜑𝑦 = 𝐸) → ((𝑦𝐵𝜓) ↔ (𝐸𝐵𝜒)))
10 riotasv2d.6 . . . . . . 7 ((𝜑𝑦 = 𝐸) → 𝐶 = 𝐹)
1110eqeq2d 2620 . . . . . 6 ((𝜑𝑦 = 𝐸) → (𝐷 = 𝐶𝐷 = 𝐹))
129, 11imbi12d 333 . . . . 5 ((𝜑𝑦 = 𝐸) → (((𝑦𝐵𝜓) → 𝐷 = 𝐶) ↔ ((𝐸𝐵𝜒) → 𝐷 = 𝐹)))
1312adantlr 747 . . . 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 33260 . . . 4 ((𝜑𝐴 ∈ V) → ((𝑦𝐵𝜓) → 𝐷 = 𝐶))
17 riotasv2d.1 . . . . 5 𝑦𝜑
18 nfv 1830 . . . . 5 𝑦 𝐴 ∈ V
1917, 18nfan 1816 . . . 4 𝑦(𝜑𝐴 ∈ V)
20 nfcvd 2752 . . . 4 ((𝜑𝐴 ∈ V) → 𝑦𝐸)
21 nfvd 1831 . . . . . . 7 (𝜑 → Ⅎ𝑦 𝐸𝐵)
22 riotasv2d.3 . . . . . . 7 (𝜑 → Ⅎ𝑦𝜒)
2321, 22nfand 1814 . . . . . 6 (𝜑 → Ⅎ𝑦(𝐸𝐵𝜒))
24 nfra1 2925 . . . . . . . . 9 𝑦𝑦𝐵 (𝜓𝑥 = 𝐶)
25 nfcv 2751 . . . . . . . . 9 𝑦𝐴
2624, 25nfriota 6520 . . . . . . . 8 𝑦(𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))
2717, 14nfceqdf 2747 . . . . . . . 8 (𝜑 → (𝑦𝐷𝑦(𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))))
2826, 27mpbiri 247 . . . . . . 7 (𝜑𝑦𝐷)
29 riotasv2d.2 . . . . . . 7 (𝜑𝑦𝐹)
3028, 29nfeqd 2758 . . . . . 6 (𝜑 → Ⅎ𝑦 𝐷 = 𝐹)
3123, 30nfimd 1812 . . . . 5 (𝜑 → Ⅎ𝑦((𝐸𝐵𝜒) → 𝐷 = 𝐹))
3231adantr 480 . . . 4 ((𝜑𝐴 ∈ V) → Ⅎ𝑦((𝐸𝐵𝜒) → 𝐷 = 𝐹))
333, 13, 16, 19, 20, 32vtocldf 3229 . . 3 ((𝜑𝐴 ∈ V) → ((𝐸𝐵𝜒) → 𝐷 = 𝐹))
343, 5, 33mp2and 711 . 2 ((𝜑𝐴 ∈ V) → 𝐷 = 𝐹)
351, 34sylan2 490 1 ((𝜑𝐴𝑉) → 𝐷 = 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738  ∀wral 2896  Vcvv 3173  ℩crio 6510 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-riotaBAD 33257 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-riota 6511  df-undef 7286 This theorem is referenced by:  riotasv2s  33262  cdleme42b  34784
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