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Theorem rexfiuz 14707
 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 3396 . . . 4 (𝑥 = ∅ → (∀𝑛𝑥 𝜑 ↔ ∀𝑛 ∈ ∅ 𝜑))
21rexralbidv 3293 . . 3 (𝑥 = ∅ → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑))
3 raleq 3396 . . 3 (𝑥 = ∅ → (∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∀𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
42, 3bibi12d 349 . 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 3396 . . . 4 (𝑥 = 𝑦 → (∀𝑛𝑥 𝜑 ↔ ∀𝑛𝑦 𝜑))
65rexralbidv 3293 . . 3 (𝑥 = 𝑦 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑))
7 raleq 3396 . . 3 (𝑥 = 𝑦 → (∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∀𝑛𝑦𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
86, 7bibi12d 349 . 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 3396 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛𝑥 𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑))
109rexralbidv 3293 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑))
11 raleq 3396 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
1210, 11bibi12d 349 . 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 3396 . . . 4 (𝑥 = 𝐴 → (∀𝑛𝑥 𝜑 ↔ ∀𝑛𝐴 𝜑))
1413rexralbidv 3293 . . 3 (𝑥 = 𝐴 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝐴 𝜑))
15 raleq 3396 . . 3 (𝑥 = 𝐴 → (∀𝑛𝑥𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∀𝑛𝐴𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
1614, 15bibi12d 349 . 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 11989 . . . . 5 0 ∈ ℤ
1817ne0ii 4286 . . . 4 ℤ ≠ ∅
19 ral0 4439 . . . . 5 𝑛 ∈ ∅ 𝜑
2019rgen2w 3146 . . . 4 𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑
21 r19.2z 4423 . . . 4 ((ℤ ≠ ∅ ∧ ∀𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑)
2218, 20, 21mp2an 691 . . 3 𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑
23 ral0 4439 . . 3 𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑
2422, 232th 267 . 2 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ ∅ 𝜑 ↔ ∀𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑)
25 anbi1 634 . . . 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 14705 . . . . 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 4153 . . . . . . 7 (∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ (∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
2827ralbii 3160 . . . . . 6 (∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∀𝑘 ∈ (ℤ𝑗)(∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
2928rexbii 3241 . . . . 5 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑))
30 ralsnsg 4593 . . . . . . . 8 (𝑧 ∈ V → (∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑[𝑧 / 𝑛]𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
31 ralcom 3345 . . . . . . . . . . 11 (∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑 ↔ ∀𝑛 ∈ {𝑧}∀𝑘 ∈ (ℤ𝑗)𝜑)
32 ralsnsg 4593 . . . . . . . . . . 11 (𝑧 ∈ V → (∀𝑛 ∈ {𝑧}∀𝑘 ∈ (ℤ𝑗)𝜑[𝑧 / 𝑛]𝑘 ∈ (ℤ𝑗)𝜑))
3331, 32syl5bb 286 . . . . . . . . . 10 (𝑧 ∈ V → (∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑[𝑧 / 𝑛]𝑘 ∈ (ℤ𝑗)𝜑))
3433rexbidv 3289 . . . . . . . . 9 (𝑧 ∈ V → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑 ↔ ∃𝑗 ∈ ℤ [𝑧 / 𝑛]𝑘 ∈ (ℤ𝑗)𝜑))
35 sbcrex 3842 . . . . . . . . 9 ([𝑧 / 𝑛]𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ [𝑧 / 𝑛]𝑘 ∈ (ℤ𝑗)𝜑)
3634, 35syl6rbbr 293 . . . . . . . 8 (𝑧 ∈ V → ([𝑧 / 𝑛]𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ {𝑧}𝜑))
3730, 36bitrd 282 . . . . . . 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 625 . . . . 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 306 . . . 4 (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)∀𝑛𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
41 ralunb 4153 . . . 4 (∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ (∀𝑛𝑦𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)𝜑))
4225, 40, 413bitr4g 317 . . 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 8755 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115   ≠ wne 3014  ∀wral 3133  ∃wrex 3134  Vcvv 3480  [wsbc 3758   ∪ cun 3917  ∅c0 4276  {csn 4550  ‘cfv 6343  Fincfn 8505  0cc0 10535  ℤcz 11978  ℤ≥cuz 12240 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-addrcl 10596  ax-rnegex 10606  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-om 7575  df-1o 8098  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-neg 10871  df-z 11979  df-uz 12241 This theorem is referenced by:  uniioombllem6  24198  rrncmslem  35218
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