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Theorem riotasv2s 34562
 Description: The value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 4904) in the form of a substitution instance. Special case of riota2f 6672. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
Hypothesis
Ref Expression
riotasv2s.2 𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
Assertion
Ref Expression
riotasv2s ((𝐴𝑉𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → 𝐷 = 𝐸 / 𝑦𝐶)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶   𝑥,𝐸,𝑦   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝐶(𝑦)   𝐷(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem riotasv2s
StepHypRef Expression
1 3simpc 1080 . 2 ((𝐴𝑉𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → (𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)))
2 simp1 1081 . 2 ((𝐴𝑉𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → 𝐴𝑉)
3 riotasv2s.2 . . . . . 6 𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
4 nfra1 2970 . . . . . . 7 𝑦𝑦𝐵 (𝜑𝑥 = 𝐶)
5 nfcv 2793 . . . . . . 7 𝑦𝐴
64, 5nfriota 6660 . . . . . 6 𝑦(𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶))
73, 6nfcxfr 2791 . . . . 5 𝑦𝐷
87nfel1 2808 . . . 4 𝑦 𝐷𝐴
9 nfv 1883 . . . . 5 𝑦 𝐸𝐵
10 nfsbc1v 3488 . . . . 5 𝑦[𝐸 / 𝑦]𝜑
119, 10nfan 1868 . . . 4 𝑦(𝐸𝐵[𝐸 / 𝑦]𝜑)
128, 11nfan 1868 . . 3 𝑦(𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑))
13 nfcsb1v 3582 . . . 4 𝑦𝐸 / 𝑦𝐶
1413a1i 11 . . 3 ((𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → 𝑦𝐸 / 𝑦𝐶)
1510a1i 11 . . 3 ((𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → Ⅎ𝑦[𝐸 / 𝑦]𝜑)
163a1i 11 . . 3 ((𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → 𝐷 = (𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
17 sbceq1a 3479 . . . 4 (𝑦 = 𝐸 → (𝜑[𝐸 / 𝑦]𝜑))
1817adantl 481 . . 3 (((𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) ∧ 𝑦 = 𝐸) → (𝜑[𝐸 / 𝑦]𝜑))
19 csbeq1a 3575 . . . 4 (𝑦 = 𝐸𝐶 = 𝐸 / 𝑦𝐶)
2019adantl 481 . . 3 (((𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) ∧ 𝑦 = 𝐸) → 𝐶 = 𝐸 / 𝑦𝐶)
21 simpl 472 . . 3 ((𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → 𝐷𝐴)
22 simprl 809 . . 3 ((𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → 𝐸𝐵)
23 simprr 811 . . 3 ((𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → [𝐸 / 𝑦]𝜑)
2412, 14, 15, 16, 18, 20, 21, 22, 23riotasv2d 34561 . 2 (((𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) ∧ 𝐴𝑉) → 𝐷 = 𝐸 / 𝑦𝐶)
251, 2, 24syl2anc 694 1 ((𝐴𝑉𝐷𝐴 ∧ (𝐸𝐵[𝐸 / 𝑦]𝜑)) → 𝐷 = 𝐸 / 𝑦𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523  Ⅎwnf 1748   ∈ wcel 2030  Ⅎwnfc 2780  ∀wral 2941  [wsbc 3468  ⦋csb 3566  ℩crio 6650 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-riotaBAD 34557 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-riota 6651  df-undef 7444 This theorem is referenced by: (None)
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