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Theorem riotasvd 36707
Description: Deduction version of riotasv 36710. (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 484 . . . . . . . 8 ((𝜑𝐴𝑉) → 𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))
3 riotasvd.2 . . . . . . . . 9 (𝜑𝐷𝐴)
43adantr 484 . . . . . . . 8 ((𝜑𝐴𝑉) → 𝐷𝐴)
52, 4eqeltrrd 2839 . . . . . . 7 ((𝜑𝐴𝑉) → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) ∈ 𝐴)
6 riotaclbgBAD 36705 . . . . . . . 8 (𝐴𝑉 → (∃!𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶) ↔ (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) ∈ 𝐴))
76adantl 485 . . . . . . 7 ((𝜑𝐴𝑉) → (∃!𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶) ↔ (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) ∈ 𝐴))
85, 7mpbird 260 . . . . . 6 ((𝜑𝐴𝑉) → ∃!𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))
9 riotasbc 7189 . . . . . 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 2741 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 = 𝐶𝑧 = 𝐶))
1211imbi2d 344 . . . . . . . 8 (𝑥 = 𝑧 → ((𝜓𝑥 = 𝐶) ↔ (𝜓𝑧 = 𝐶)))
1312ralbidv 3118 . . . . . . 7 (𝑥 = 𝑧 → (∀𝑦𝐵 (𝜓𝑥 = 𝐶) ↔ ∀𝑦𝐵 (𝜓𝑧 = 𝐶)))
14 nfra1 3140 . . . . . . . . . 10 𝑦𝑦𝐵 (𝜓𝑥 = 𝐶)
15 nfcv 2904 . . . . . . . . . 10 𝑦𝐴
1614, 15nfriota 7183 . . . . . . . . 9 𝑦(𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))
1716nfeq2 2921 . . . . . . . 8 𝑦 𝑧 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))
18 eqeq1 2741 . . . . . . . . 9 (𝑧 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) → (𝑧 = 𝐶 ↔ (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶))
1918imbi2d 344 . . . . . . . 8 (𝑧 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) → ((𝜓𝑧 = 𝐶) ↔ (𝜓 → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶)))
2017, 19ralbid 3154 . . . . . . 7 (𝑧 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) → (∀𝑦𝐵 (𝜓𝑧 = 𝐶) ↔ ∀𝑦𝐵 (𝜓 → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶)))
2113, 20sbcie2g 3736 . . . . . 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 235 . . . 4 ((𝜑𝐴𝑉) → ∀𝑦𝐵 (𝜓 → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶))
24 rsp 3127 . . . 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 414 . 2 ((𝜑𝐴𝑉) → ((𝑦𝐵𝜓) → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶))
272eqeq1d 2739 . 2 ((𝜑𝐴𝑉) → (𝐷 = 𝐶 ↔ (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶))
2826, 27sylibrd 262 1 ((𝜑𝐴𝑉) → ((𝑦𝐵𝜓) → 𝐷 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wral 3061  ∃!wreu 3063  [wsbc 3694  crio 7169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-riotaBAD 36704
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-riota 7170  df-undef 8015
This theorem is referenced by:  riotasv2d  36708  riotasv  36710  riotasv3d  36711  cdleme32a  38192
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