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Theorem riotasv2d 37448
Description: Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5363). Special case of riota2f 7343. (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 3466 . 2 (𝐴𝑉𝐴 ∈ V)
2 riotasv2d.8 . . . 4 (𝜑𝐸𝐵)
32adantr 482 . . 3 ((𝜑𝐴 ∈ V) → 𝐸𝐵)
4 riotasv2d.9 . . . 4 (𝜑𝜒)
54adantr 482 . . 3 ((𝜑𝐴 ∈ V) → 𝜒)
6 eleq1 2826 . . . . . . . 8 (𝑦 = 𝐸 → (𝑦𝐵𝐸𝐵))
76adantl 483 . . . . . . 7 ((𝜑𝑦 = 𝐸) → (𝑦𝐵𝐸𝐵))
8 riotasv2d.5 . . . . . . 7 ((𝜑𝑦 = 𝐸) → (𝜓𝜒))
97, 8anbi12d 632 . . . . . 6 ((𝜑𝑦 = 𝐸) → ((𝑦𝐵𝜓) ↔ (𝐸𝐵𝜒)))
10 riotasv2d.6 . . . . . . 7 ((𝜑𝑦 = 𝐸) → 𝐶 = 𝐹)
1110eqeq2d 2748 . . . . . 6 ((𝜑𝑦 = 𝐸) → (𝐷 = 𝐶𝐷 = 𝐹))
129, 11imbi12d 345 . . . . 5 ((𝜑𝑦 = 𝐸) → (((𝑦𝐵𝜓) → 𝐷 = 𝐶) ↔ ((𝐸𝐵𝜒) → 𝐷 = 𝐹)))
1312adantlr 714 . . . 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 37447 . . . 4 ((𝜑𝐴 ∈ V) → ((𝑦𝐵𝜓) → 𝐷 = 𝐶))
17 riotasv2d.1 . . . . 5 𝑦𝜑
18 nfv 1918 . . . . 5 𝑦 𝐴 ∈ V
1917, 18nfan 1903 . . . 4 𝑦(𝜑𝐴 ∈ V)
20 nfcvd 2909 . . . 4 ((𝜑𝐴 ∈ V) → 𝑦𝐸)
21 nfvd 1919 . . . . . . 7 (𝜑 → Ⅎ𝑦 𝐸𝐵)
22 riotasv2d.3 . . . . . . 7 (𝜑 → Ⅎ𝑦𝜒)
2321, 22nfand 1901 . . . . . 6 (𝜑 → Ⅎ𝑦(𝐸𝐵𝜒))
24 nfra1 3270 . . . . . . . . 9 𝑦𝑦𝐵 (𝜓𝑥 = 𝐶)
25 nfcv 2908 . . . . . . . . 9 𝑦𝐴
2624, 25nfriota 7331 . . . . . . . 8 𝑦(𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))
2717, 14nfceqdf 2903 . . . . . . . 8 (𝜑 → (𝑦𝐷𝑦(𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))))
2826, 27mpbiri 258 . . . . . . 7 (𝜑𝑦𝐷)
29 riotasv2d.2 . . . . . . 7 (𝜑𝑦𝐹)
3028, 29nfeqd 2918 . . . . . 6 (𝜑 → Ⅎ𝑦 𝐷 = 𝐹)
3123, 30nfimd 1898 . . . . 5 (𝜑 → Ⅎ𝑦((𝐸𝐵𝜒) → 𝐷 = 𝐹))
3231adantr 482 . . . 4 ((𝜑𝐴 ∈ V) → Ⅎ𝑦((𝐸𝐵𝜒) → 𝐷 = 𝐹))
333, 13, 16, 19, 20, 32vtocldf 3513 . . 3 ((𝜑𝐴 ∈ V) → ((𝐸𝐵𝜒) → 𝐷 = 𝐹))
343, 5, 33mp2and 698 . 2 ((𝜑𝐴 ∈ V) → 𝐷 = 𝐹)
351, 34sylan2 594 1 ((𝜑𝐴𝑉) → 𝐷 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wnf 1786  wcel 2107  wnfc 2888  wral 3065  Vcvv 3448  crio 7317
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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-riotaBAD 37444
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-riota 7318  df-undef 8209
This theorem is referenced by:  riotasv2s  37449  cdleme42b  38970
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