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Theorem riotasv 38323
Description: Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5392). Special case of riota2f 7383. (Contributed by NM, 26-Jan-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
Hypotheses
Ref Expression
riotasv.1 𝐴 ∈ V
riotasv.2 𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
Assertion
Ref Expression
riotasv ((𝐷𝐴𝑦𝐵𝜑) → 𝐷 = 𝐶)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝐵(𝑦)   𝐶(𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem riotasv
StepHypRef Expression
1 riotasv.1 . . 3 𝐴 ∈ V
2 riotasv.2 . . . . 5 𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
32a1i 11 . . . 4 (𝐷𝐴𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
4 id 22 . . . 4 (𝐷𝐴𝐷𝐴)
53, 4riotasvd 38320 . . 3 ((𝐷𝐴𝐴 ∈ V) → ((𝑦𝐵𝜑) → 𝐷 = 𝐶))
61, 5mpan2 688 . 2 (𝐷𝐴 → ((𝑦𝐵𝜑) → 𝐷 = 𝐶))
763impib 1113 1 ((𝐷𝐴𝑦𝐵𝜑) → 𝐷 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3053  Vcvv 3466  crio 7357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-riotaBAD 38317
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-riota 7358  df-undef 8254
This theorem is referenced by:  cdleme26e  39724  cdleme26eALTN  39726  cdleme26fALTN  39727  cdleme26f  39728  cdleme26f2ALTN  39729  cdleme26f2  39730
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