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Theorem rexanuz2nf 45443
Description: A simple counterexample related to theorem rexanuz2 15385, demonstrating the necessity of its disjoint variable constraints. Here, 𝑗 appears free in 𝜑, showing that without these constraints, rexanuz2 15385 and similar theorems would not hold (see rexanre 15382 and rexanuz 15381). (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 12539 . . . . . . . 8 0 ∈ ℕ0
2 nn0ge0 12549 . . . . . . . . 9 (𝑘 ∈ ℕ0 → 0 ≤ 𝑘)
32rgen 3061 . . . . . . . 8 𝑘 ∈ ℕ0 0 ≤ 𝑘
4 fveq2 6907 . . . . . . . . . . . 12 (𝑗 = 0 → (ℤ𝑗) = (ℤ‘0))
5 nn0uz 12918 . . . . . . . . . . . 12 0 = (ℤ‘0)
64, 5eqtr4di 2793 . . . . . . . . . . 11 (𝑗 = 0 → (ℤ𝑗) = ℕ0)
76raleqdv 3324 . . . . . . . . . 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 5170 . . . . . . . . . . . . 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 3174 . . . . . . . . . 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 2903 . . . . . . . . 9 𝑗0
2018, 19nfcxfr 2901 . . . . . . . 8 𝑗𝑍
2120, 19, 18rexeqif 45109 . . . . . . 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 3091 . . . . . . 7 (∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∀𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘))
2524rexbii 3092 . . . . . 6 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘))
2622, 25mpbir 231 . . . . 5 𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑
27 1nn0 12540 . . . . . . . 8 1 ∈ ℕ0
28 nngt0 12295 . . . . . . . . 9 (𝑘 ∈ ℕ → 0 < 𝑘)
2928rgen 3061 . . . . . . . 8 𝑘 ∈ ℕ 0 < 𝑘
30 fveq2 6907 . . . . . . . . . . 11 (𝑗 = 1 → (ℤ𝑗) = (ℤ‘1))
31 nnuz 12919 . . . . . . . . . . 11 ℕ = (ℤ‘1)
3230, 31eqtr4di 2793 . . . . . . . . . 10 (𝑗 = 1 → (ℤ𝑗) = ℕ)
3332raleqdv 3324 . . . . . . . . 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 45109 . . . . . . 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 3091 . . . . . . 7 (∀𝑘 ∈ (ℤ𝑗)𝜓 ↔ ∀𝑘 ∈ (ℤ𝑗)0 < 𝑘)
4039rexbii 3092 . . . . . 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 1912 . . . . . . . . 9 𝑘 ¬ ((𝑗 = 0 ∧ 𝑗𝑗) ∧ 0 < 𝑗)
44 nfcv 2903 . . . . . . . . 9 𝑘𝑗
45 nfcv 2903 . . . . . . . . 9 𝑘(ℤ𝑗)
465uzid3 45385 . . . . . . . . . 10 (𝑗 ∈ ℕ0𝑗 ∈ (ℤ𝑗))
4746adantr 480 . . . . . . . . 9 ((𝑗 ∈ ℕ0𝑗 = 0) → 𝑗 ∈ (ℤ𝑗))
48 0re 11261 . . . . . . . . . . . . . 14 0 ∈ ℝ
4948ltnri 11368 . . . . . . . . . . . . 13 ¬ 0 < 0
5049a1i 11 . . . . . . . . . . . 12 (𝑗 = 0 → ¬ 0 < 0)
51 eqcom 2742 . . . . . . . . . . . . 13 (𝑗 = 0 ↔ 0 = 𝑗)
5251biimpi 216 . . . . . . . . . . . 12 (𝑗 = 0 → 0 = 𝑗)
5350, 52brneqtrd 45016 . . . . . . . . . . 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 5152 . . . . . . . . . . . . 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 5152 . . . . . . . . . . . 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 45110 . . . . . . . 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 45110 . . . . . . . 8 ((𝑗 ∈ ℕ0 ∧ ¬ 𝑗 = 0) → ∃𝑘 ∈ (ℤ𝑗) ¬ (𝜑𝜓))
7063, 69pm2.61dan 813 . . . . . . 7 (𝑗 ∈ ℕ0 → ∃𝑘 ∈ (ℤ𝑗) ¬ (𝜑𝜓))
71 rexnal 3098 . . . . . . 7 (∃𝑘 ∈ (ℤ𝑗) ¬ (𝜑𝜓) ↔ ¬ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
7270, 71sylib 218 . . . . . 6 (𝑗 ∈ ℕ0 → ¬ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
7372nrex 3072 . . . . 5 ¬ ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)(𝜑𝜓)
7420, 19, 18rexeqif 45109 . . . . 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 45107 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 1537  wcel 2106  wral 3059  wrex 3068   class class class wbr 5148  cfv 6563  0cc0 11153  1c1 11154   < clt 11293  cle 11294  cn 12264  0cn0 12524  cuz 12876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877
This theorem is referenced by: (None)
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