Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  riotasvd Structured version   Visualization version   GIF version

Theorem riotasvd 35574
Description: Deduction version of riotasv 35577. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
riotasvd.1 (𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))
riotasvd.2 (𝜑𝐷𝐴)
Assertion
Ref Expression
riotasvd ((𝜑𝐴𝑉) → ((𝑦𝐵𝜓) → 𝐷 = 𝐶))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵   𝑥,𝐶   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝐵(𝑦)   𝐶(𝑦)   𝐷(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem riotasvd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 riotasvd.1 . . . . . . . . 9 (𝜑𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))
21adantr 473 . . . . . . . 8 ((𝜑𝐴𝑉) → 𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))
3 riotasvd.2 . . . . . . . . 9 (𝜑𝐷𝐴)
43adantr 473 . . . . . . . 8 ((𝜑𝐴𝑉) → 𝐷𝐴)
52, 4eqeltrrd 2860 . . . . . . 7 ((𝜑𝐴𝑉) → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) ∈ 𝐴)
6 riotaclbgBAD 35572 . . . . . . . 8 (𝐴𝑉 → (∃!𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶) ↔ (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) ∈ 𝐴))
76adantl 474 . . . . . . 7 ((𝜑𝐴𝑉) → (∃!𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶) ↔ (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) ∈ 𝐴))
85, 7mpbird 249 . . . . . 6 ((𝜑𝐴𝑉) → ∃!𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))
9 riotasbc 6950 . . . . . 6 (∃!𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶) → [(𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) / 𝑥]𝑦𝐵 (𝜓𝑥 = 𝐶))
108, 9syl 17 . . . . 5 ((𝜑𝐴𝑉) → [(𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) / 𝑥]𝑦𝐵 (𝜓𝑥 = 𝐶))
11 eqeq1 2775 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 = 𝐶𝑧 = 𝐶))
1211imbi2d 333 . . . . . . . 8 (𝑥 = 𝑧 → ((𝜓𝑥 = 𝐶) ↔ (𝜓𝑧 = 𝐶)))
1312ralbidv 3140 . . . . . . 7 (𝑥 = 𝑧 → (∀𝑦𝐵 (𝜓𝑥 = 𝐶) ↔ ∀𝑦𝐵 (𝜓𝑧 = 𝐶)))
14 nfra1 3162 . . . . . . . . . 10 𝑦𝑦𝐵 (𝜓𝑥 = 𝐶)
15 nfcv 2925 . . . . . . . . . 10 𝑦𝐴
1614, 15nfriota 6944 . . . . . . . . 9 𝑦(𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))
1716nfeq2 2940 . . . . . . . 8 𝑦 𝑧 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))
18 eqeq1 2775 . . . . . . . . 9 (𝑧 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) → (𝑧 = 𝐶 ↔ (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶))
1918imbi2d 333 . . . . . . . 8 (𝑧 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) → ((𝜓𝑧 = 𝐶) ↔ (𝜓 → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶)))
2017, 19ralbid 3171 . . . . . . 7 (𝑧 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) → (∀𝑦𝐵 (𝜓𝑧 = 𝐶) ↔ ∀𝑦𝐵 (𝜓 → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶)))
2113, 20sbcie2g 3708 . . . . . 6 ((𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) ∈ 𝐴 → ([(𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) / 𝑥]𝑦𝐵 (𝜓𝑥 = 𝐶) ↔ ∀𝑦𝐵 (𝜓 → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶)))
225, 21syl 17 . . . . 5 ((𝜑𝐴𝑉) → ([(𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) / 𝑥]𝑦𝐵 (𝜓𝑥 = 𝐶) ↔ ∀𝑦𝐵 (𝜓 → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶)))
2310, 22mpbid 224 . . . 4 ((𝜑𝐴𝑉) → ∀𝑦𝐵 (𝜓 → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶))
24 rsp 3148 . . . 4 (∀𝑦𝐵 (𝜓 → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶) → (𝑦𝐵 → (𝜓 → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶)))
2523, 24syl 17 . . 3 ((𝜑𝐴𝑉) → (𝑦𝐵 → (𝜓 → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶)))
2625impd 402 . 2 ((𝜑𝐴𝑉) → ((𝑦𝐵𝜓) → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶))
272eqeq1d 2773 . 2 ((𝜑𝐴𝑉) → (𝐷 = 𝐶 ↔ (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶))
2826, 27sylibrd 251 1 ((𝜑𝐴𝑉) → ((𝑦𝐵𝜓) → 𝐷 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1508  wcel 2051  wral 3081  ∃!wreu 3083  [wsbc 3674  crio 6934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-riotaBAD 35571
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-iota 6149  df-fun 6187  df-fv 6193  df-riota 6935  df-undef 7740
This theorem is referenced by:  riotasv2d  35575  riotasv  35577  riotasv3d  35578  cdleme32a  37059
  Copyright terms: Public domain W3C validator