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Theorem rexfiuz 15292
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 3314 . . . 4 (𝑥 = ∅ → (∀𝑛𝑥 𝜑 ↔ ∀𝑛 ∈ ∅ 𝜑))
21rexralbidv 3212 . . 3 (𝑥 = ∅ → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑))
3 raleq 3314 . . 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 3314 . . . 4 (𝑥 = 𝑦 → (∀𝑛𝑥 𝜑 ↔ ∀𝑛𝑦 𝜑))
65rexralbidv 3212 . . 3 (𝑥 = 𝑦 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑))
7 raleq 3314 . . 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 3314 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛𝑥 𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑))
109rexralbidv 3212 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑))
11 raleq 3314 . . 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 3314 . . . 4 (𝑥 = 𝐴 → (∀𝑛𝑥 𝜑 ↔ ∀𝑛𝐴 𝜑))
1413rexralbidv 3212 . . 3 (𝑥 = 𝐴 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝐴 𝜑))
15 raleq 3314 . . 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 12567 . . . . 5 0 ∈ ℤ
1817ne0ii 4330 . . . 4 ℤ ≠ ∅
19 ral0 4505 . . . . 5 𝑛 ∈ ∅ 𝜑
2019rgen2w 3058 . . . 4 𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑
21 r19.2z 4487 . . . 4 ((ℤ ≠ ∅ ∧ ∀𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑)
2218, 20, 21mp2an 689 . . 3 𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑
23 ral0 4505 . . 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 631 . . . 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 15290 . . . . 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 4184 . . . . . . 7 (∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ (∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
2827ralbii 3085 . . . . . 6 (∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∀𝑘 ∈ (ℤ𝑗)(∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
2928rexbii 3086 . . . . 5 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
30 ralsnsg 4665 . . . . . . . 8 (𝑧 ∈ V → (∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑[𝑧 / 𝑛]𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
31 sbcrex 3862 . . . . . . . . 9 ([𝑧 / 𝑛]𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ [𝑧 / 𝑛]𝑘 ∈ (ℤ𝑗)𝜑)
32 ralcom 3278 . . . . . . . . . . 11 (∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑 ↔ ∀𝑛 ∈ {𝑧}∀𝑘 ∈ (ℤ𝑗)𝜑)
33 ralsnsg 4665 . . . . . . . . . . 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 3170 . . . . . . . . 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 3472 . . . . . 6 (∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑)
3938anbi2i 622 . . . . 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 4184 . . . 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 9161 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 205  wa 395   = wceq 1533  wcel 2098  wne 2932  wral 3053  wrex 3062  Vcvv 3466  [wsbc 3770  cun 3939  c0 4315  {csn 4621  cfv 6534  Fincfn 8936  0cc0 11107  cz 12556  cuz 12820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-addrcl 11168  ax-rnegex 11178  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-om 7850  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-neg 11445  df-z 12557  df-uz 12821
This theorem is referenced by:  uniioombllem6  25441  rrncmslem  37194
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