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Theorem riotasvd 34745
Description: Deduction version of riotasv 34748. (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 472 . . . . . . . 8 ((𝜑𝐴𝑉) → 𝐷 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)))
3 riotasvd.2 . . . . . . . . 9 (𝜑𝐷𝐴)
43adantr 472 . . . . . . . 8 ((𝜑𝐴𝑉) → 𝐷𝐴)
52, 4eqeltrrd 2840 . . . . . . 7 ((𝜑𝐴𝑉) → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) ∈ 𝐴)
6 riotaclbgBAD 34743 . . . . . . . 8 (𝐴𝑉 → (∃!𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶) ↔ (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) ∈ 𝐴))
76adantl 473 . . . . . . 7 ((𝜑𝐴𝑉) → (∃!𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶) ↔ (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) ∈ 𝐴))
85, 7mpbird 247 . . . . . 6 ((𝜑𝐴𝑉) → ∃!𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))
9 riotasbc 6789 . . . . . 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 2764 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 = 𝐶𝑧 = 𝐶))
1211imbi2d 329 . . . . . . . 8 (𝑥 = 𝑧 → ((𝜓𝑥 = 𝐶) ↔ (𝜓𝑧 = 𝐶)))
1312ralbidv 3124 . . . . . . 7 (𝑥 = 𝑧 → (∀𝑦𝐵 (𝜓𝑥 = 𝐶) ↔ ∀𝑦𝐵 (𝜓𝑧 = 𝐶)))
14 nfra1 3079 . . . . . . . . . 10 𝑦𝑦𝐵 (𝜓𝑥 = 𝐶)
15 nfcv 2902 . . . . . . . . . 10 𝑦𝐴
1614, 15nfriota 6783 . . . . . . . . 9 𝑦(𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))
1716nfeq2 2918 . . . . . . . 8 𝑦 𝑧 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶))
18 eqeq1 2764 . . . . . . . . 9 (𝑧 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) → (𝑧 = 𝐶 ↔ (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶))
1918imbi2d 329 . . . . . . . 8 (𝑧 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) → ((𝜓𝑧 = 𝐶) ↔ (𝜓 → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶)))
2017, 19ralbid 3121 . . . . . . 7 (𝑧 = (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) → (∀𝑦𝐵 (𝜓𝑧 = 𝐶) ↔ ∀𝑦𝐵 (𝜓 → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶)))
2113, 20sbcie2g 3610 . . . . . 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 222 . . . 4 ((𝜑𝐴𝑉) → ∀𝑦𝐵 (𝜓 → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶))
24 rsp 3067 . . . 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 446 . 2 ((𝜑𝐴𝑉) → ((𝑦𝐵𝜓) → (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶))
272eqeq1d 2762 . 2 ((𝜑𝐴𝑉) → (𝐷 = 𝐶 ↔ (𝑥𝐴𝑦𝐵 (𝜓𝑥 = 𝐶)) = 𝐶))
2826, 27sylibrd 249 1 ((𝜑𝐴𝑉) → ((𝑦𝐵𝜓) → 𝐷 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  ∃!wreu 3052  [wsbc 3576  crio 6773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-riotaBAD 34742
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-riota 6774  df-undef 7568
This theorem is referenced by:  riotasv2d  34746  riotasv  34748  riotasv3d  34749  cdleme32a  36231
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