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Theorem riotasv2d 33061
Description: Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 4792). Special case of riota2f 6507. (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 3181 . 2 (𝐴𝑉𝐴 ∈ V)
2 riotasv2d.8 . . . 4 (𝜑𝐸𝐵)
32adantr 479 . . 3 ((𝜑𝐴 ∈ V) → 𝐸𝐵)
4 riotasv2d.9 . . . 4 (𝜑𝜒)
54adantr 479 . . 3 ((𝜑𝐴 ∈ V) → 𝜒)
6 eleq1 2672 . . . . . . . 8 (𝑦 = 𝐸 → (𝑦𝐵𝐸𝐵))
76adantl 480 . . . . . . 7 ((𝜑𝑦 = 𝐸) → (𝑦𝐵𝐸𝐵))
8 riotasv2d.5 . . . . . . 7 ((𝜑𝑦 = 𝐸) → (𝜓𝜒))
97, 8anbi12d 742 . . . . . 6 ((𝜑𝑦 = 𝐸) → ((𝑦𝐵𝜓) ↔ (𝐸𝐵𝜒)))
10 riotasv2d.6 . . . . . . 7 ((𝜑𝑦 = 𝐸) → 𝐶 = 𝐹)
1110eqeq2d 2616 . . . . . 6 ((𝜑𝑦 = 𝐸) → (𝐷 = 𝐶𝐷 = 𝐹))
129, 11imbi12d 332 . . . . 5 ((𝜑𝑦 = 𝐸) → (((𝑦𝐵𝜓) → 𝐷 = 𝐶) ↔ ((𝐸𝐵𝜒) → 𝐷 = 𝐹)))
1312adantlr 746 . . . 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 33060 . . . 4 ((𝜑𝐴 ∈ V) → ((𝑦𝐵𝜓) → 𝐷 = 𝐶))
17 riotasv2d.1 . . . . 5 𝑦𝜑
18 nfv 1829 . . . . 5 𝑦 𝐴 ∈ V
1917, 18nfan 1815 . . . 4 𝑦(𝜑𝐴 ∈ V)
20 nfcvd 2748 . . . 4 ((𝜑𝐴 ∈ V) → 𝑦𝐸)
21 nfvd 1830 . . . . . . 7 (𝜑 → Ⅎ𝑦 𝐸𝐵)
22 riotasv2d.3 . . . . . . 7 (𝜑 → Ⅎ𝑦𝜒)
2321, 22nfand 1813 . . . . . 6 (𝜑 → Ⅎ𝑦(𝐸𝐵𝜒))
24 nfra1 2921 . . . . . . . . 9 𝑦𝑦𝐵 (𝜓𝑥 = 𝐶)
25 nfcv 2747 . . . . . . . . 9 𝑦𝐴
2624, 25nfriota 6495 . . . . . . . 8 𝑦(𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))
2717, 14nfceqdf 2743 . . . . . . . 8 (𝜑 → (𝑦𝐷𝑦(𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))))
2826, 27mpbiri 246 . . . . . . 7 (𝜑𝑦𝐷)
29 riotasv2d.2 . . . . . . 7 (𝜑𝑦𝐹)
3028, 29nfeqd 2754 . . . . . 6 (𝜑 → Ⅎ𝑦 𝐷 = 𝐹)
3123, 30nfimd 1811 . . . . 5 (𝜑 → Ⅎ𝑦((𝐸𝐵𝜒) → 𝐷 = 𝐹))
3231adantr 479 . . . 4 ((𝜑𝐴 ∈ V) → Ⅎ𝑦((𝐸𝐵𝜒) → 𝐷 = 𝐹))
333, 13, 16, 19, 20, 32vtocldf 3225 . . 3 ((𝜑𝐴 ∈ V) → ((𝐸𝐵𝜒) → 𝐷 = 𝐹))
343, 5, 33mp2and 710 . 2 ((𝜑𝐴 ∈ V) → 𝐷 = 𝐹)
351, 34sylan2 489 1 ((𝜑𝐴𝑉) → 𝐷 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wnf 1698  wcel 1976  wnfc 2734  wral 2892  Vcvv 3169  crio 6485
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 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-riotaBAD 33057
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 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-iota 5751  df-fun 5789  df-fv 5795  df-riota 6486  df-undef 7260
This theorem is referenced by:  riotasv2s  33062  cdleme42b  34584
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