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Theorem riotasvd 37447
Description: Deduction version of riotasv 37450. (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 482 . . . . . . . 8 ((𝜑𝐴𝑉) → 𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))
3 riotasvd.2 . . . . . . . . 9 (𝜑𝐷𝐴)
43adantr 482 . . . . . . . 8 ((𝜑𝐴𝑉) → 𝐷𝐴)
52, 4eqeltrrd 2839 . . . . . . 7 ((𝜑𝐴𝑉) → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) ∈ 𝐴)
6 riotaclbgBAD 37445 . . . . . . . 8 (𝐴𝑉 → (∃!𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶) ↔ (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) ∈ 𝐴))
76adantl 483 . . . . . . 7 ((𝜑𝐴𝑉) → (∃!𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶) ↔ (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) ∈ 𝐴))
85, 7mpbird 257 . . . . . 6 ((𝜑𝐴𝑉) → ∃!𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))
9 riotasbc 7337 . . . . . 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 341 . . . . . . . 8 (𝑥 = 𝑧 → ((𝜓𝑥 = 𝐶) ↔ (𝜓𝑧 = 𝐶)))
1312ralbidv 3175 . . . . . . 7 (𝑥 = 𝑧 → (∀𝑦𝐵 (𝜓𝑥 = 𝐶) ↔ ∀𝑦𝐵 (𝜓𝑧 = 𝐶)))
14 nfra1 3270 . . . . . . . . . 10 𝑦𝑦𝐵 (𝜓𝑥 = 𝐶)
15 nfcv 2908 . . . . . . . . . 10 𝑦𝐴
1614, 15nfriota 7331 . . . . . . . . 9 𝑦(𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))
1716nfeq2 2925 . . . . . . . 8 𝑦 𝑧 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))
18 eqeq1 2741 . . . . . . . . 9 (𝑧 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) → (𝑧 = 𝐶 ↔ (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶))
1918imbi2d 341 . . . . . . . 8 (𝑧 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) → ((𝜓𝑧 = 𝐶) ↔ (𝜓 → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶)))
2017, 19ralbid 3259 . . . . . . 7 (𝑧 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) → (∀𝑦𝐵 (𝜓𝑧 = 𝐶) ↔ ∀𝑦𝐵 (𝜓 → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶)))
2113, 20sbcie2g 3786 . . . . . 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 231 . . . 4 ((𝜑𝐴𝑉) → ∀𝑦𝐵 (𝜓 → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶))
24 rsp 3233 . . . 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 412 . 2 ((𝜑𝐴𝑉) → ((𝑦𝐵𝜓) → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶))
272eqeq1d 2739 . 2 ((𝜑𝐴𝑉) → (𝐷 = 𝐶 ↔ (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶))
2826, 27sylibrd 259 1 ((𝜑𝐴𝑉) → ((𝑦𝐵𝜓) → 𝐷 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3065  ∃!wreu 3354  [wsbc 3744  crio 7317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-riotaBAD 37444
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-riota 7318  df-undef 8209
This theorem is referenced by:  riotasv2d  37448  riotasv  37450  riotasv3d  37451  cdleme32a  38933
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