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Theorem riotasv 35488
 Description: Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5151). Special case of riota2f 6952. (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 35485 . . 3 ((𝐷𝐴𝐴 ∈ V) → ((𝑦𝐵𝜑) → 𝐷 = 𝐶))
61, 5mpan2 678 . 2 (𝐷𝐴 → ((𝑦𝐵𝜑) → 𝐷 = 𝐶))
763impib 1096 1 ((𝐷𝐴𝑦𝐵𝜑) → 𝐷 = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2048  ∀wral 3082  Vcvv 3409  ℩crio 6930 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-riotaBAD 35482 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-iota 6146  df-fun 6184  df-fv 6190  df-riota 6931  df-undef 7735 This theorem is referenced by:  cdleme26e  36888  cdleme26eALTN  36890  cdleme26fALTN  36891  cdleme26f  36892  cdleme26f2ALTN  36893  cdleme26f2  36894
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