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Theorem riotasv 38940
Description: Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5408). Special case of riota2f 7411. (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 38937 . . 3 ((𝐷𝐴𝐴 ∈ V) → ((𝑦𝐵𝜑) → 𝐷 = 𝐶))
61, 5mpan2 691 . 2 (𝐷𝐴 → ((𝑦𝐵𝜑) → 𝐷 = 𝐶))
763impib 1115 1 ((𝐷𝐴𝑦𝐵𝜑) → 𝐷 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  Vcvv 3477  crio 7386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-riotaBAD 38934
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-riota 7387  df-undef 8296
This theorem is referenced by:  cdleme26e  40341  cdleme26eALTN  40343  cdleme26fALTN  40344  cdleme26f  40345  cdleme26f2ALTN  40346  cdleme26f2  40347
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