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Theorem rexanuz2nf 44937
Description: A simple counterexample related to theorem rexanuz2 15326, demonstrating the necessity of its disjoint variable constraints. Here, 𝑗 appears free in 𝜑, showing that without these constraints, rexanuz2 15326 and similar theorems would not hold (see rexanre 15323 and rexanuz 15322). (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 12515 . . . . . . . 8 0 ∈ ℕ0
2 nn0ge0 12525 . . . . . . . . 9 (𝑘 ∈ ℕ0 → 0 ≤ 𝑘)
32rgen 3053 . . . . . . . 8 𝑘 ∈ ℕ0 0 ≤ 𝑘
4 fveq2 6891 . . . . . . . . . . . 12 (𝑗 = 0 → (ℤ𝑗) = (ℤ‘0))
5 nn0uz 12892 . . . . . . . . . . . 12 0 = (ℤ‘0)
64, 5eqtr4di 2783 . . . . . . . . . . 11 (𝑗 = 0 → (ℤ𝑗) = ℕ0)
76raleqdv 3315 . . . . . . . . . 10 (𝑗 = 0 → (∀𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘) ↔ ∀𝑘 ∈ ℕ0 (𝑗 = 0 ∧ 𝑗𝑘)))
82ad2antlr 725 . . . . . . . . . . . 12 (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ (𝑗 = 0 ∧ 𝑗𝑘)) → 0 ≤ 𝑘)
9 simpll 765 . . . . . . . . . . . . 13 (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → 𝑗 = 0)
10 simpr 483 . . . . . . . . . . . . . 14 (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → 0 ≤ 𝑘)
119, 10eqbrtrd 5165 . . . . . . . . . . . . 13 (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → 𝑗𝑘)
129, 11jca 510 . . . . . . . . . . . 12 (((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) ∧ 0 ≤ 𝑘) → (𝑗 = 0 ∧ 𝑗𝑘))
138, 12impbida 799 . . . . . . . . . . 11 ((𝑗 = 0 ∧ 𝑘 ∈ ℕ0) → ((𝑗 = 0 ∧ 𝑗𝑘) ↔ 0 ≤ 𝑘))
1413ralbidva 3166 . . . . . . . . . 10 (𝑗 = 0 → (∀𝑘 ∈ ℕ0 (𝑗 = 0 ∧ 𝑗𝑘) ↔ ∀𝑘 ∈ ℕ0 0 ≤ 𝑘))
157, 14bitrd 278 . . . . . . . . 9 (𝑗 = 0 → (∀𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘) ↔ ∀𝑘 ∈ ℕ0 0 ≤ 𝑘))
1615rspcev 3602 . . . . . . . 8 ((0 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 0 ≤ 𝑘) → ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘))
171, 3, 16mp2an 690 . . . . . . 7 𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘)
18 rexanuz2nf.1 . . . . . . . . 9 𝑍 = ℕ0
19 nfcv 2892 . . . . . . . . 9 𝑗0
2018, 19nfcxfr 2890 . . . . . . . 8 𝑗𝑍
2120, 19, 18rexeqif 44601 . . . . . . 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 3083 . . . . . . 7 (∀𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∀𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘))
2524rexbii 3084 . . . . . 6 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑗 = 0 ∧ 𝑗𝑘))
2622, 25mpbir 230 . . . . 5 𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑
27 1nn0 12516 . . . . . . . 8 1 ∈ ℕ0
28 nngt0 12271 . . . . . . . . 9 (𝑘 ∈ ℕ → 0 < 𝑘)
2928rgen 3053 . . . . . . . 8 𝑘 ∈ ℕ 0 < 𝑘
30 fveq2 6891 . . . . . . . . . . 11 (𝑗 = 1 → (ℤ𝑗) = (ℤ‘1))
31 nnuz 12893 . . . . . . . . . . 11 ℕ = (ℤ‘1)
3230, 31eqtr4di 2783 . . . . . . . . . 10 (𝑗 = 1 → (ℤ𝑗) = ℕ)
3332raleqdv 3315 . . . . . . . . 9 (𝑗 = 1 → (∀𝑘 ∈ (ℤ𝑗)0 < 𝑘 ↔ ∀𝑘 ∈ ℕ 0 < 𝑘))
3433rspcev 3602 . . . . . . . 8 ((1 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ 0 < 𝑘) → ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)0 < 𝑘)
3527, 29, 34mp2an 690 . . . . . . 7 𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)0 < 𝑘
3620, 19, 18rexeqif 44601 . . . . . . 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 3083 . . . . . . 7 (∀𝑘 ∈ (ℤ𝑗)𝜓 ↔ ∀𝑘 ∈ (ℤ𝑗)0 < 𝑘)
4039rexbii 3084 . . . . . 6 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)0 < 𝑘)
4137, 40mpbir 230 . . . . 5 𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓
4226, 41pm3.2i 469 . . . 4 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓)
43 nfv 1909 . . . . . . . . 9 𝑘 ¬ ((𝑗 = 0 ∧ 𝑗𝑗) ∧ 0 < 𝑗)
44 nfcv 2892 . . . . . . . . 9 𝑘𝑗
45 nfcv 2892 . . . . . . . . 9 𝑘(ℤ𝑗)
465uzid3 44879 . . . . . . . . . 10 (𝑗 ∈ ℕ0𝑗 ∈ (ℤ𝑗))
4746adantr 479 . . . . . . . . 9 ((𝑗 ∈ ℕ0𝑗 = 0) → 𝑗 ∈ (ℤ𝑗))
48 0re 11244 . . . . . . . . . . . . . 14 0 ∈ ℝ
4948ltnri 11351 . . . . . . . . . . . . 13 ¬ 0 < 0
5049a1i 11 . . . . . . . . . . . 12 (𝑗 = 0 → ¬ 0 < 0)
51 eqcom 2732 . . . . . . . . . . . . 13 (𝑗 = 0 ↔ 0 = 𝑗)
5251biimpi 215 . . . . . . . . . . . 12 (𝑗 = 0 → 0 = 𝑗)
5350, 52brneqtrd 44506 . . . . . . . . . . 11 (𝑗 = 0 → ¬ 0 < 𝑗)
5453intnand 487 . . . . . . . . . 10 (𝑗 = 0 → ¬ ((𝑗 = 0 ∧ 𝑗𝑗) ∧ 0 < 𝑗))
5554adantl 480 . . . . . . . . 9 ((𝑗 ∈ ℕ0𝑗 = 0) → ¬ ((𝑗 = 0 ∧ 𝑗𝑗) ∧ 0 < 𝑗))
56 breq2 5147 . . . . . . . . . . . . 13 (𝑘 = 𝑗 → (𝑗𝑘𝑗𝑗))
5756anbi2d 628 . . . . . . . . . . . 12 (𝑘 = 𝑗 → ((𝑗 = 0 ∧ 𝑗𝑘) ↔ (𝑗 = 0 ∧ 𝑗𝑗)))
5823, 57bitrid 282 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝜑 ↔ (𝑗 = 0 ∧ 𝑗𝑗)))
59 breq2 5147 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (0 < 𝑘 ↔ 0 < 𝑗))
6038, 59bitrid 282 . . . . . . . . . . 11 (𝑘 = 𝑗 → (𝜓 ↔ 0 < 𝑗))
6158, 60anbi12d 630 . . . . . . . . . 10 (𝑘 = 𝑗 → ((𝜑𝜓) ↔ ((𝑗 = 0 ∧ 𝑗𝑗) ∧ 0 < 𝑗)))
6261notbid 317 . . . . . . . . 9 (𝑘 = 𝑗 → (¬ (𝜑𝜓) ↔ ¬ ((𝑗 = 0 ∧ 𝑗𝑗) ∧ 0 < 𝑗)))
6343, 44, 45, 47, 55, 62rspced 44602 . . . . . . . 8 ((𝑗 ∈ ℕ0𝑗 = 0) → ∃𝑘 ∈ (ℤ𝑗) ¬ (𝜑𝜓))
6446adantr 479 . . . . . . . . 9 ((𝑗 ∈ ℕ0 ∧ ¬ 𝑗 = 0) → 𝑗 ∈ (ℤ𝑗))
65 id 22 . . . . . . . . . . . 12 𝑗 = 0 → ¬ 𝑗 = 0)
6665intnanrd 488 . . . . . . . . . . 11 𝑗 = 0 → ¬ (𝑗 = 0 ∧ 𝑗𝑗))
6766intnanrd 488 . . . . . . . . . 10 𝑗 = 0 → ¬ ((𝑗 = 0 ∧ 𝑗𝑗) ∧ 0 < 𝑗))
6867adantl 480 . . . . . . . . 9 ((𝑗 ∈ ℕ0 ∧ ¬ 𝑗 = 0) → ¬ ((𝑗 = 0 ∧ 𝑗𝑗) ∧ 0 < 𝑗))
6943, 44, 45, 64, 68, 62rspced 44602 . . . . . . . 8 ((𝑗 ∈ ℕ0 ∧ ¬ 𝑗 = 0) → ∃𝑘 ∈ (ℤ𝑗) ¬ (𝜑𝜓))
7063, 69pm2.61dan 811 . . . . . . 7 (𝑗 ∈ ℕ0 → ∃𝑘 ∈ (ℤ𝑗) ¬ (𝜑𝜓))
71 rexnal 3090 . . . . . . 7 (∃𝑘 ∈ (ℤ𝑗) ¬ (𝜑𝜓) ↔ ¬ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
7270, 71sylib 217 . . . . . 6 (𝑗 ∈ ℕ0 → ¬ ∀𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
7372nrex 3064 . . . . 5 ¬ ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)(𝜑𝜓)
7420, 19, 18rexeqif 44601 . . . . 5 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜑𝜓) ↔ ∃𝑗 ∈ ℕ0𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
7573, 74mtbir 322 . . . 4 ¬ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜑𝜓)
7642, 75pm3.2i 469 . . 3 ((∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜑 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓) ∧ ¬ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝜑𝜓))
77 annim 402 . . 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 44599 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 394   = wceq 1533  wcel 2098  wral 3051  wrex 3060   class class class wbr 5143  cfv 6542  0cc0 11136  1c1 11137   < clt 11276  cle 11277  cn 12240  0cn0 12500  cuz 12850
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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-er 8721  df-en 8961  df-dom 8962  df-sdom 8963  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-n0 12501  df-z 12587  df-uz 12851
This theorem is referenced by: (None)
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