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Theorem rexanuz2nf 45503
Description: A simple counterexample related to theorem rexanuz2 15388, demonstrating the necessity of its disjoint variable constraints. Here, 𝑗 appears free in 𝜑, showing that without these constraints, rexanuz2 15388 and similar theorems would not hold (see rexanre 15385 and rexanuz 15384). (Contributed by Glauco Siliprandi, 15-Feb-2025.)
Hypotheses
Ref Expression
rexanuz2nf.1 𝑍 = ℕ0
rexanuz2nf.2 (𝜑 ↔ (𝑗 = 0 ∧ 𝑗𝑘))
rexanuz2nf.3 (𝜓 ↔ 0 < 𝑘)
Assertion
Ref Expression
rexanuz2nf ¬ (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓))
Distinct variable group:   𝑗,𝑘
Allowed substitution hints:   𝜑(𝑗,𝑘)   𝜓(𝑗,𝑘)   𝑍(𝑗,𝑘)

Proof of Theorem rexanuz2nf
StepHypRef Expression
1 0nn0 12541 . . . . . . . 8 0 ∈ ℕ0
2 nn0ge0 12551 . . . . . . . . 9 (𝑘 ∈ ℕ0 → 0 ≤ 𝑘)
32rgen 3063 . . . . . . . 8 𝑘 ∈ ℕ0 0 ≤ 𝑘
4 fveq2 6906 . . . . . . . . . . . 12 (𝑗 = 0 → (ℤ𝑗) = (ℤ‘0))
5 nn0uz 12920 . . . . . . . . . . . 12 0 = (ℤ‘0)
64, 5eqtr4di 2795 . . . . . . . . . . 11 (𝑗 = 0 → (ℤ𝑗) = ℕ0)
76raleqdv 3326 . . . . . . . . . 10 (𝑗 = 0 → (∀𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘) ↔ ∀𝑘 ∈ ℕ0 (𝑗 = 0 ∧ 𝑗𝑘)))
82ad2antlr 727 . . . . . . . . . . . 12 (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ (𝑗 = 0 ∧ 𝑗𝑘)) → 0 ≤ 𝑘)
9 simpll 767 . . . . . . . . . . . . 13 (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → 𝑗 = 0)
10 simpr 484 . . . . . . . . . . . . . 14 (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → 0 ≤ 𝑘)
119, 10eqbrtrd 5165 . . . . . . . . . . . . 13 (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → 𝑗𝑘)
129, 11jca 511 . . . . . . . . . . . 12 (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → (𝑗 = 0 ∧ 𝑗𝑘))
138, 12impbida 801 . . . . . . . . . . 11 ((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) → ((𝑗 = 0 ∧ 𝑗𝑘) ↔ 0 ≤ 𝑘))
1413ralbidva 3176 . . . . . . . . . 10 (𝑗 = 0 → (∀𝑘 ∈ ℕ0 (𝑗 = 0 ∧ 𝑗𝑘) ↔ ∀𝑘 ∈ ℕ0 0 ≤ 𝑘))
157, 14bitrd 279 . . . . . . . . 9 (𝑗 = 0 → (∀𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘) ↔ ∀𝑘 ∈ ℕ0 0 ≤ 𝑘))
1615rspcev 3622 . . . . . . . 8 ((0 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 0 ≤ 𝑘) → ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘))
171, 3, 16mp2an 692 . . . . . . 7 𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘)
18 rexanuz2nf.1 . . . . . . . . 9 𝑍 = ℕ0
19 nfcv 2905 . . . . . . . . 9 𝑗0
2018, 19nfcxfr 2903 . . . . . . . 8 𝑗𝑍
2120, 19, 18rexeqif 45171 . . . . . . 7 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘) ↔ ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘))
2217, 21mpbir 231 . . . . . 6 𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘)
23 rexanuz2nf.2 . . . . . . . 8 (𝜑 ↔ (𝑗 = 0 ∧ 𝑗𝑘))
2423ralbii 3093 . . . . . . 7 (∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∀𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘))
2524rexbii 3094 . . . . . 6 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘))
2622, 25mpbir 231 . . . . 5 𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑
27 1nn0 12542 . . . . . . . 8 1 ∈ ℕ0
28 nngt0 12297 . . . . . . . . 9 (𝑘 ∈ ℕ → 0 < 𝑘)
2928rgen 3063 . . . . . . . 8 𝑘 ∈ ℕ 0 < 𝑘
30 fveq2 6906 . . . . . . . . . . 11 (𝑗 = 1 → (ℤ𝑗) = (ℤ‘1))
31 nnuz 12921 . . . . . . . . . . 11 ℕ = (ℤ‘1)
3230, 31eqtr4di 2795 . . . . . . . . . 10 (𝑗 = 1 → (ℤ𝑗) = ℕ)
3332raleqdv 3326 . . . . . . . . 9 (𝑗 = 1 → (∀𝑘 ∈ (ℤ𝑗)0 < 𝑘 ↔ ∀𝑘 ∈ ℕ 0 < 𝑘))
3433rspcev 3622 . . . . . . . 8 ((1 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ 0 < 𝑘) → ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)0 < 𝑘)
3527, 29, 34mp2an 692 . . . . . . 7 𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)0 < 𝑘
3620, 19, 18rexeqif 45171 . . . . . . 7 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)0 < 𝑘 ↔ ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)0 < 𝑘)
3735, 36mpbir 231 . . . . . 6 𝑗𝑍𝑘 ∈ (ℤ𝑗)0 < 𝑘
38 rexanuz2nf.3 . . . . . . . 8 (𝜓 ↔ 0 < 𝑘)
3938ralbii 3093 . . . . . . 7 (∀𝑘 ∈ (ℤ𝑗)𝜓 ↔ ∀𝑘 ∈ (ℤ𝑗)0 < 𝑘)
4039rexbii 3094 . . . . . 6 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)0 < 𝑘)
4137, 40mpbir 231 . . . . 5 𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓
4226, 41pm3.2i 470 . . . 4 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓)
43 nfv 1914 . . . . . . . . 9 𝑘 ¬ ((𝑗 = 0 ∧ 𝑗𝑗) ∧ 0 < 𝑗)
44 nfcv 2905 . . . . . . . . 9 𝑘𝑗
45 nfcv 2905 . . . . . . . . 9 𝑘(ℤ𝑗)
465uzid3 45446 . . . . . . . . . 10 (𝑗 ∈ ℕ0𝑗 ∈ (ℤ𝑗))
4746adantr 480 . . . . . . . . 9 ((𝑗 ∈ ℕ0𝑗 = 0) → 𝑗 ∈ (ℤ𝑗))
48 0re 11263 . . . . . . . . . . . . . 14 0 ∈ ℝ
4948ltnri 11370 . . . . . . . . . . . . 13 ¬ 0 < 0
5049a1i 11 . . . . . . . . . . . 12 (𝑗 = 0 → ¬ 0 < 0)
51 eqcom 2744 . . . . . . . . . . . . 13 (𝑗 = 0 ↔ 0 = 𝑗)
5251biimpi 216 . . . . . . . . . . . 12 (𝑗 = 0 → 0 = 𝑗)
5350, 52brneqtrd 45081 . . . . . . . . . . 11 (𝑗 = 0 → ¬ 0 < 𝑗)
5453intnand 488 . . . . . . . . . 10 (𝑗 = 0 → ¬ ((𝑗 = 0 ∧ 𝑗𝑗) ∧ 0 < 𝑗))
5554adantl 481 . . . . . . . . 9 ((𝑗 ∈ ℕ0𝑗 = 0) → ¬ ((𝑗 = 0 ∧ 𝑗𝑗) ∧ 0 < 𝑗))
56 breq2 5147 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → (𝑗𝑘𝑗𝑗))
5756anbi2d 630 . . . . . . . . . . . 12 (𝑘 = 𝑗 → ((𝑗 = 0 ∧ 𝑗𝑘) ↔ (𝑗 = 0 ∧ 𝑗𝑗)))
5823, 57bitrid 283 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝜑 ↔ (𝑗 = 0 ∧ 𝑗𝑗)))
59 breq2 5147 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (0 < 𝑘 ↔ 0 < 𝑗))
6038, 59bitrid 283 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝜓 ↔ 0 < 𝑗))
6158, 60anbi12d 632 . . . . . . . . . 10 (𝑘 = 𝑗 → ((𝜑𝜓) ↔ ((𝑗 = 0 ∧ 𝑗𝑗) ∧ 0 < 𝑗)))
6261notbid 318 . . . . . . . . 9 (𝑘 = 𝑗 → (¬ (𝜑𝜓) ↔ ¬ ((𝑗 = 0 ∧ 𝑗𝑗) ∧ 0 < 𝑗)))
6343, 44, 45, 47, 55, 62rspced 45172 . . . . . . . 8 ((𝑗 ∈ ℕ0𝑗 = 0) → ∃𝑘 ∈ (ℤ𝑗) ¬ (𝜑𝜓))
6446adantr 480 . . . . . . . . 9 ((𝑗 ∈ ℕ0 ∧ ¬ 𝑗 = 0) → 𝑗 ∈ (ℤ𝑗))
65 id 22 . . . . . . . . . . . 12 𝑗 = 0 → ¬ 𝑗 = 0)
6665intnanrd 489 . . . . . . . . . . 11 𝑗 = 0 → ¬ (𝑗 = 0 ∧ 𝑗𝑗))
6766intnanrd 489 . . . . . . . . . 10 𝑗 = 0 → ¬ ((𝑗 = 0 ∧ 𝑗𝑗) ∧ 0 < 𝑗))
6867adantl 481 . . . . . . . . 9 ((𝑗 ∈ ℕ0 ∧ ¬ 𝑗 = 0) → ¬ ((𝑗 = 0 ∧ 𝑗𝑗) ∧ 0 < 𝑗))
6943, 44, 45, 64, 68, 62rspced 45172 . . . . . . . 8 ((𝑗 ∈ ℕ0 ∧ ¬ 𝑗 = 0) → ∃𝑘 ∈ (ℤ𝑗) ¬ (𝜑𝜓))
7063, 69pm2.61dan 813 . . . . . . 7 (𝑗 ∈ ℕ0 → ∃𝑘 ∈ (ℤ𝑗) ¬ (𝜑𝜓))
71 rexnal 3100 . . . . . . 7 (∃𝑘 ∈ (ℤ𝑗) ¬ (𝜑𝜓) ↔ ¬ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
7270, 71sylib 218 . . . . . 6 (𝑗 ∈ ℕ0 → ¬ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
7372nrex 3074 . . . . 5 ¬ ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)(𝜑𝜓)
7420, 19, 18rexeqif 45171 . . . . 5 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
7573, 74mtbir 323 . . . 4 ¬ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜑𝜓)
7642, 75pm3.2i 470 . . 3 ((∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓) ∧ ¬ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
77 annim 403 . . 3 (((∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓) ∧ ¬ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜑𝜓)) ↔ ¬ ((∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜑𝜓)))
7876, 77mpbi 230 . 2 ¬ ((∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
7978nimnbi2 45169 1 ¬ (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070   class class class wbr 5143  cfv 6561  0cc0 11155  1c1 11156   < clt 11295  cle 11296  cn 12266  0cn0 12526  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-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
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-iun 4993  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-pred 6321  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-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879
This theorem is referenced by: (None)
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