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Theorem rexfiuz 15386
Description: Combine finitely many different upper integer properties into one. (Contributed by Mario Carneiro, 6-Jun-2014.)
Assertion
Ref Expression
rexfiuz (𝐴 ∈ Fin → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝐴 𝜑 ↔ ∀𝑛𝐴𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
Distinct variable groups:   𝑗,𝑘,𝑛,𝐴   𝜑,𝑗
Allowed substitution hints:   𝜑(𝑘,𝑛)

Proof of Theorem rexfiuz
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 raleq 3323 . . . 4 (𝑥 = ∅ → (∀𝑛𝑥 𝜑 ↔ ∀𝑛 ∈ ∅ 𝜑))
21rexralbidv 3223 . . 3 (𝑥 = ∅ → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑))
3 raleq 3323 . . 3 (𝑥 = ∅ → (∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∀𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
42, 3bibi12d 345 . 2 (𝑥 = ∅ → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑 ↔ ∀𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑)))
5 raleq 3323 . . . 4 (𝑥 = 𝑦 → (∀𝑛𝑥 𝜑 ↔ ∀𝑛𝑦 𝜑))
65rexralbidv 3223 . . 3 (𝑥 = 𝑦 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑))
7 raleq 3323 . . 3 (𝑥 = 𝑦 → (∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∀𝑛𝑦𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
86, 7bibi12d 345 . 2 (𝑥 = 𝑦 → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑 ↔ ∀𝑛𝑦𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑)))
9 raleq 3323 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛𝑥 𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑))
109rexralbidv 3223 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑))
11 raleq 3323 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
1210, 11bibi12d 345 . 2 (𝑥 = (𝑦 ∪ {𝑧}) → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑)))
13 raleq 3323 . . . 4 (𝑥 = 𝐴 → (∀𝑛𝑥 𝜑 ↔ ∀𝑛𝐴 𝜑))
1413rexralbidv 3223 . . 3 (𝑥 = 𝐴 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝐴 𝜑))
15 raleq 3323 . . 3 (𝑥 = 𝐴 → (∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∀𝑛𝐴𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
1614, 15bibi12d 345 . 2 (𝑥 = 𝐴 → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝐴 𝜑 ↔ ∀𝑛𝐴𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑)))
17 0z 12624 . . . . 5 0 ∈ ℤ
1817ne0ii 4344 . . . 4 ℤ ≠ ∅
19 ral0 4513 . . . . 5 𝑛 ∈ ∅ 𝜑
2019rgen2w 3066 . . . 4 𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑
21 r19.2z 4495 . . . 4 ((ℤ ≠ ∅ ∧ ∀𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑)
2218, 20, 21mp2an 692 . . 3 𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑
23 ral0 4513 . . 3 𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑
2422, 232th 264 . 2 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑 ↔ ∀𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑)
25 anbi1 633 . . . 4 ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑 ↔ ∀𝑛𝑦𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑) → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑) ↔ (∀𝑛𝑦𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑)))
26 rexanuz 15384 . . . . 5 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑))
27 ralunb 4197 . . . . . . 7 (∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ (∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
2827ralbii 3093 . . . . . 6 (∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∀𝑘 ∈ (ℤ𝑗)(∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
2928rexbii 3094 . . . . 5 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
30 ralsnsg 4670 . . . . . . . 8 (𝑧 ∈ V → (∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑[𝑧 / 𝑛]𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
31 sbcrex 3875 . . . . . . . . 9 ([𝑧 / 𝑛]𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ [𝑧 / 𝑛]𝑘 ∈ (ℤ𝑗)𝜑)
32 ralcom 3289 . . . . . . . . . . 11 (∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑 ↔ ∀𝑛 ∈ {𝑧}∀𝑘 ∈ (ℤ𝑗)𝜑)
33 ralsnsg 4670 . . . . . . . . . . 11 (𝑧 ∈ V → (∀𝑛 ∈ {𝑧}∀𝑘 ∈ (ℤ𝑗)𝜑[𝑧 / 𝑛]𝑘 ∈ (ℤ𝑗)𝜑))
3432, 33bitrid 283 . . . . . . . . . 10 (𝑧 ∈ V → (∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑[𝑧 / 𝑛]𝑘 ∈ (ℤ𝑗)𝜑))
3534rexbidv 3179 . . . . . . . . 9 (𝑧 ∈ V → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑 ↔ ∃𝑗 ∈ ℤ [𝑧 / 𝑛]𝑘 ∈ (ℤ𝑗)𝜑))
3631, 35bitr4id 290 . . . . . . . 8 (𝑧 ∈ V → ([𝑧 / 𝑛]𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑))
3730, 36bitrd 279 . . . . . . 7 (𝑧 ∈ V → (∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑))
3837elv 3485 . . . . . 6 (∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑)
3938anbi2i 623 . . . . 5 ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑))
4026, 29, 393bitr4i 303 . . . 4 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
41 ralunb 4197 . . . 4 (∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ (∀𝑛𝑦𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
4225, 40, 413bitr4g 314 . . 3 ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑 ↔ ∀𝑛𝑦𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
4342a1i 11 . 2 (𝑦 ∈ Fin → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑 ↔ ∀𝑛𝑦𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑)))
444, 8, 12, 16, 24, 43findcard2 9204 1 (𝐴 ∈ Fin → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝐴 𝜑 ↔ ∀𝑛𝐴𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  Vcvv 3480  [wsbc 3788  cun 3949  c0 4333  {csn 4626  cfv 6561  Fincfn 8985  0cc0 11155  cz 12613  cuz 12878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-addrcl 11216  ax-rnegex 11226  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-neg 11495  df-z 12614  df-uz 12879
This theorem is referenced by:  uniioombllem6  25623  rrncmslem  37839
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