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Theorem riotasv2d 36125
Description: Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5276). Special case of riota2f 7111. (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 3488 . 2 (𝐴𝑉𝐴 ∈ V)
2 riotasv2d.8 . . . 4 (𝜑𝐸𝐵)
32adantr 483 . . 3 ((𝜑𝐴 ∈ V) → 𝐸𝐵)
4 riotasv2d.9 . . . 4 (𝜑𝜒)
54adantr 483 . . 3 ((𝜑𝐴 ∈ V) → 𝜒)
6 eleq1 2898 . . . . . . . 8 (𝑦 = 𝐸 → (𝑦𝐵𝐸𝐵))
76adantl 484 . . . . . . 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 2831 . . . . . 6 ((𝜑𝑦 = 𝐸) → (𝐷 = 𝐶𝐷 = 𝐹))
129, 11imbi12d 347 . . . . 5 ((𝜑𝑦 = 𝐸) → (((𝑦𝐵𝜓) → 𝐷 = 𝐶) ↔ ((𝐸𝐵𝜒) → 𝐷 = 𝐹)))
1312adantlr 713 . . . 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 36124 . . . 4 ((𝜑𝐴 ∈ V) → ((𝑦𝐵𝜓) → 𝐷 = 𝐶))
17 riotasv2d.1 . . . . 5 𝑦𝜑
18 nfv 1915 . . . . 5 𝑦 𝐴 ∈ V
1917, 18nfan 1900 . . . 4 𝑦(𝜑𝐴 ∈ V)
20 nfcvd 2974 . . . 4 ((𝜑𝐴 ∈ V) → 𝑦𝐸)
21 nfvd 1916 . . . . . . 7 (𝜑 → Ⅎ𝑦 𝐸𝐵)
22 riotasv2d.3 . . . . . . 7 (𝜑 → Ⅎ𝑦𝜒)
2321, 22nfand 1898 . . . . . 6 (𝜑 → Ⅎ𝑦(𝐸𝐵𝜒))
24 nfra1 3206 . . . . . . . . 9 𝑦𝑦𝐵 (𝜓𝑥 = 𝐶)
25 nfcv 2973 . . . . . . . . 9 𝑦𝐴
2624, 25nfriota 7099 . . . . . . . 8 𝑦(𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))
2717, 14nfceqdf 2968 . . . . . . . 8 (𝜑 → (𝑦𝐷𝑦(𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))))
2826, 27mpbiri 260 . . . . . . 7 (𝜑𝑦𝐷)
29 riotasv2d.2 . . . . . . 7 (𝜑𝑦𝐹)
3028, 29nfeqd 2983 . . . . . 6 (𝜑 → Ⅎ𝑦 𝐷 = 𝐹)
3123, 30nfimd 1895 . . . . 5 (𝜑 → Ⅎ𝑦((𝐸𝐵𝜒) → 𝐷 = 𝐹))
3231adantr 483 . . . 4 ((𝜑𝐴 ∈ V) → Ⅎ𝑦((𝐸𝐵𝜒) → 𝐷 = 𝐹))
333, 13, 16, 19, 20, 32vtocldf 3531 . . 3 ((𝜑𝐴 ∈ V) → ((𝐸𝐵𝜒) → 𝐷 = 𝐹))
343, 5, 33mp2and 697 . 2 ((𝜑𝐴 ∈ V) → 𝐷 = 𝐹)
351, 34sylan2 594 1 ((𝜑𝐴𝑉) → 𝐷 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wnf 1784  wcel 2114  wnfc 2957  wral 3125  Vcvv 3470  crio 7086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435  ax-riotaBAD 36121
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-iota 6286  df-fun 6329  df-fv 6335  df-riota 7087  df-undef 7913
This theorem is referenced by:  riotasv2s  36126  cdleme42b  37646
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