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Theorem rexanuz2nf 44780
Description: A simple counterexample related to theorem rexanuz2 15302, demonstrating the necessity of its disjoint variable constraints. Here, 𝑗 appears free in 𝜑, showing that without these constraints, rexanuz2 15302 and similar theorems would not hold (see rexanre 15299 and rexanuz 15298). (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 12491 . . . . . . . 8 0 ∈ ℕ0
2 nn0ge0 12501 . . . . . . . . 9 (𝑘 ∈ ℕ0 → 0 ≤ 𝑘)
32rgen 3057 . . . . . . . 8 𝑘 ∈ ℕ0 0 ≤ 𝑘
4 fveq2 6885 . . . . . . . . . . . 12 (𝑗 = 0 → (ℤ𝑗) = (ℤ‘0))
5 nn0uz 12868 . . . . . . . . . . . 12 0 = (ℤ‘0)
64, 5eqtr4di 2784 . . . . . . . . . . 11 (𝑗 = 0 → (ℤ𝑗) = ℕ0)
76raleqdv 3319 . . . . . . . . . 10 (𝑗 = 0 → (∀𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘) ↔ ∀𝑘 ∈ ℕ0 (𝑗 = 0 ∧ 𝑗𝑘)))
82ad2antlr 724 . . . . . . . . . . . 12 (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ (𝑗 = 0 ∧ 𝑗𝑘)) → 0 ≤ 𝑘)
9 simpll 764 . . . . . . . . . . . . 13 (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → 𝑗 = 0)
10 simpr 484 . . . . . . . . . . . . . 14 (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → 0 ≤ 𝑘)
119, 10eqbrtrd 5163 . . . . . . . . . . . . 13 (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → 𝑗𝑘)
129, 11jca 511 . . . . . . . . . . . 12 (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → (𝑗 = 0 ∧ 𝑗𝑘))
138, 12impbida 798 . . . . . . . . . . 11 ((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) → ((𝑗 = 0 ∧ 𝑗𝑘) ↔ 0 ≤ 𝑘))
1413ralbidva 3169 . . . . . . . . . 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 3606 . . . . . . . 8 ((0 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 0 ≤ 𝑘) → ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘))
171, 3, 16mp2an 689 . . . . . . 7 𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘)
18 rexanuz2nf.1 . . . . . . . . 9 𝑍 = ℕ0
19 nfcv 2897 . . . . . . . . 9 𝑗0
2018, 19nfcxfr 2895 . . . . . . . 8 𝑗𝑍
2120, 19, 18rexeqif 44441 . . . . . . 7 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘) ↔ ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘))
2217, 21mpbir 230 . . . . . 6 𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘)
23 rexanuz2nf.2 . . . . . . . 8 (𝜑 ↔ (𝑗 = 0 ∧ 𝑗𝑘))
2423ralbii 3087 . . . . . . 7 (∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∀𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘))
2524rexbii 3088 . . . . . 6 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘))
2622, 25mpbir 230 . . . . 5 𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑
27 1nn0 12492 . . . . . . . 8 1 ∈ ℕ0
28 nngt0 12247 . . . . . . . . 9 (𝑘 ∈ ℕ → 0 < 𝑘)
2928rgen 3057 . . . . . . . 8 𝑘 ∈ ℕ 0 < 𝑘
30 fveq2 6885 . . . . . . . . . . 11 (𝑗 = 1 → (ℤ𝑗) = (ℤ‘1))
31 nnuz 12869 . . . . . . . . . . 11 ℕ = (ℤ‘1)
3230, 31eqtr4di 2784 . . . . . . . . . 10 (𝑗 = 1 → (ℤ𝑗) = ℕ)
3332raleqdv 3319 . . . . . . . . 9 (𝑗 = 1 → (∀𝑘 ∈ (ℤ𝑗)0 < 𝑘 ↔ ∀𝑘 ∈ ℕ 0 < 𝑘))
3433rspcev 3606 . . . . . . . 8 ((1 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ 0 < 𝑘) → ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)0 < 𝑘)
3527, 29, 34mp2an 689 . . . . . . 7 𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)0 < 𝑘
3620, 19, 18rexeqif 44441 . . . . . . 7 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)0 < 𝑘 ↔ ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)0 < 𝑘)
3735, 36mpbir 230 . . . . . 6 𝑗𝑍𝑘 ∈ (ℤ𝑗)0 < 𝑘
38 rexanuz2nf.3 . . . . . . . 8 (𝜓 ↔ 0 < 𝑘)
3938ralbii 3087 . . . . . . 7 (∀𝑘 ∈ (ℤ𝑗)𝜓 ↔ ∀𝑘 ∈ (ℤ𝑗)0 < 𝑘)
4039rexbii 3088 . . . . . 6 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)0 < 𝑘)
4137, 40mpbir 230 . . . . 5 𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓
4226, 41pm3.2i 470 . . . 4 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓)
43 nfv 1909 . . . . . . . . 9 𝑘 ¬ ((𝑗 = 0 ∧ 𝑗𝑗) ∧ 0 < 𝑗)
44 nfcv 2897 . . . . . . . . 9 𝑘𝑗
45 nfcv 2897 . . . . . . . . 9 𝑘(ℤ𝑗)
465uzid3 44722 . . . . . . . . . 10 (𝑗 ∈ ℕ0𝑗 ∈ (ℤ𝑗))
4746adantr 480 . . . . . . . . 9 ((𝑗 ∈ ℕ0𝑗 = 0) → 𝑗 ∈ (ℤ𝑗))
48 0re 11220 . . . . . . . . . . . . . 14 0 ∈ ℝ
4948ltnri 11327 . . . . . . . . . . . . 13 ¬ 0 < 0
5049a1i 11 . . . . . . . . . . . 12 (𝑗 = 0 → ¬ 0 < 0)
51 eqcom 2733 . . . . . . . . . . . . 13 (𝑗 = 0 ↔ 0 = 𝑗)
5251biimpi 215 . . . . . . . . . . . 12 (𝑗 = 0 → 0 = 𝑗)
5350, 52brneqtrd 44345 . . . . . . . . . . 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 5145 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → (𝑗𝑘𝑗𝑗))
5756anbi2d 628 . . . . . . . . . . . 12 (𝑘 = 𝑗 → ((𝑗 = 0 ∧ 𝑗𝑘) ↔ (𝑗 = 0 ∧ 𝑗𝑗)))
5823, 57bitrid 283 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝜑 ↔ (𝑗 = 0 ∧ 𝑗𝑗)))
59 breq2 5145 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (0 < 𝑘 ↔ 0 < 𝑗))
6038, 59bitrid 283 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝜓 ↔ 0 < 𝑗))
6158, 60anbi12d 630 . . . . . . . . . 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 44442 . . . . . . . 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 44442 . . . . . . . 8 ((𝑗 ∈ ℕ0 ∧ ¬ 𝑗 = 0) → ∃𝑘 ∈ (ℤ𝑗) ¬ (𝜑𝜓))
7063, 69pm2.61dan 810 . . . . . . 7 (𝑗 ∈ ℕ0 → ∃𝑘 ∈ (ℤ𝑗) ¬ (𝜑𝜓))
71 rexnal 3094 . . . . . . 7 (∃𝑘 ∈ (ℤ𝑗) ¬ (𝜑𝜓) ↔ ¬ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
7270, 71sylib 217 . . . . . 6 (𝑗 ∈ ℕ0 → ¬ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
7372nrex 3068 . . . . 5 ¬ ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)(𝜑𝜓)
7420, 19, 18rexeqif 44441 . . . . 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 229 . 2 ¬ ((∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
7978nimnbi2 44439 1 ¬ (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wral 3055  wrex 3064   class class class wbr 5141  cfv 6537  0cc0 11112  1c1 11113   < clt 11252  cle 11253  cn 12216  0cn0 12476  cuz 12826
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 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827
This theorem is referenced by: (None)
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