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Theorem rexfiuz 14464
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 3350 . . . 4 (𝑥 = ∅ → (∀𝑛𝑥 𝜑 ↔ ∀𝑛 ∈ ∅ 𝜑))
21rexralbidv 3268 . . 3 (𝑥 = ∅ → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑))
3 raleq 3350 . . 3 (𝑥 = ∅ → (∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∀𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
42, 3bibi12d 337 . 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 3350 . . . 4 (𝑥 = 𝑦 → (∀𝑛𝑥 𝜑 ↔ ∀𝑛𝑦 𝜑))
65rexralbidv 3268 . . 3 (𝑥 = 𝑦 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑))
7 raleq 3350 . . 3 (𝑥 = 𝑦 → (∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∀𝑛𝑦𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
86, 7bibi12d 337 . 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 3350 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛𝑥 𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑))
109rexralbidv 3268 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑))
11 raleq 3350 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
1210, 11bibi12d 337 . 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 3350 . . . 4 (𝑥 = 𝐴 → (∀𝑛𝑥 𝜑 ↔ ∀𝑛𝐴 𝜑))
1413rexralbidv 3268 . . 3 (𝑥 = 𝐴 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝐴 𝜑))
15 raleq 3350 . . 3 (𝑥 = 𝐴 → (∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∀𝑛𝐴𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
1614, 15bibi12d 337 . 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 11715 . . . . 5 0 ∈ ℤ
1817ne0ii 4153 . . . 4 ℤ ≠ ∅
19 ral0 4298 . . . . 5 𝑛 ∈ ∅ 𝜑
2019rgen2w 3134 . . . 4 𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑
21 r19.2z 4282 . . . 4 ((ℤ ≠ ∅ ∧ ∀𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑)
2218, 20, 21mp2an 685 . . 3 𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑
23 ral0 4298 . . 3 𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑
2422, 232th 256 . 2 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑 ↔ ∀𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑)
25 anbi1 627 . . . 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 14462 . . . . 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 4021 . . . . . . 7 (∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ (∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
2827ralbii 3189 . . . . . 6 (∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∀𝑘 ∈ (ℤ𝑗)(∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
2928rexbii 3251 . . . . 5 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
30 vex 3417 . . . . . . 7 𝑧 ∈ V
31 ralsnsg 4436 . . . . . . . 8 (𝑧 ∈ V → (∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑[𝑧 / 𝑛]𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
32 ralcom 3308 . . . . . . . . . . 11 (∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑 ↔ ∀𝑛 ∈ {𝑧}∀𝑘 ∈ (ℤ𝑗)𝜑)
33 ralsnsg 4436 . . . . . . . . . . 11 (𝑧 ∈ V → (∀𝑛 ∈ {𝑧}∀𝑘 ∈ (ℤ𝑗)𝜑[𝑧 / 𝑛]𝑘 ∈ (ℤ𝑗)𝜑))
3432, 33syl5bb 275 . . . . . . . . . 10 (𝑧 ∈ V → (∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑[𝑧 / 𝑛]𝑘 ∈ (ℤ𝑗)𝜑))
3534rexbidv 3262 . . . . . . . . 9 (𝑧 ∈ V → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑 ↔ ∃𝑗 ∈ ℤ [𝑧 / 𝑛]𝑘 ∈ (ℤ𝑗)𝜑))
36 sbcrex 3738 . . . . . . . . 9 ([𝑧 / 𝑛]𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ [𝑧 / 𝑛]𝑘 ∈ (ℤ𝑗)𝜑)
3735, 36syl6rbbr 282 . . . . . . . 8 (𝑧 ∈ V → ([𝑧 / 𝑛]𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑))
3831, 37bitrd 271 . . . . . . 7 (𝑧 ∈ V → (∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑))
3930, 38ax-mp 5 . . . . . 6 (∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑)
4039anbi2i 618 . . . . 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))
4126, 29, 403bitr4i 295 . . . 4 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
42 ralunb 4021 . . . 4 (∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ (∀𝑛𝑦𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
4325, 41, 423bitr4g 306 . . 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))
4443a1i 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)))
454, 8, 12, 16, 24, 44findcard2 8469 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 198  wa 386   = wceq 1658  wcel 2166  wne 2999  wral 3117  wrex 3118  Vcvv 3414  [wsbc 3662  cun 3796  c0 4144  {csn 4397  cfv 6123  Fincfn 8222  0cc0 10252  cz 11704  cuz 11968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-addrcl 10313  ax-rnegex 10323  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-om 7327  df-1o 7826  df-er 8009  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-neg 10588  df-z 11705  df-uz 11969
This theorem is referenced by:  uniioombllem6  23754  rrncmslem  34173
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