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Theorem indstr 9050
Description: Strong Mathematical Induction for positive integers (inference schema). (Contributed by NM, 17-Aug-2001.)
Hypotheses
Ref Expression
indstr.1 (𝑥 = 𝑦 → (𝜑𝜓))
indstr.2 (𝑥 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) → 𝜑))
Assertion
Ref Expression
indstr (𝑥 ∈ ℕ → 𝜑)
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem indstr
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 3841 . . . . 5 (𝑧 = 1 → (𝑦 < 𝑧𝑦 < 1))
21imbi1d 229 . . . 4 (𝑧 = 1 → ((𝑦 < 𝑧𝜓) ↔ (𝑦 < 1 → 𝜓)))
32ralbidv 2380 . . 3 (𝑧 = 1 → (∀𝑦 ∈ ℕ (𝑦 < 𝑧𝜓) ↔ ∀𝑦 ∈ ℕ (𝑦 < 1 → 𝜓)))
4 breq2 3841 . . . . 5 (𝑧 = 𝑤 → (𝑦 < 𝑧𝑦 < 𝑤))
54imbi1d 229 . . . 4 (𝑧 = 𝑤 → ((𝑦 < 𝑧𝜓) ↔ (𝑦 < 𝑤𝜓)))
65ralbidv 2380 . . 3 (𝑧 = 𝑤 → (∀𝑦 ∈ ℕ (𝑦 < 𝑧𝜓) ↔ ∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓)))
7 breq2 3841 . . . . 5 (𝑧 = (𝑤 + 1) → (𝑦 < 𝑧𝑦 < (𝑤 + 1)))
87imbi1d 229 . . . 4 (𝑧 = (𝑤 + 1) → ((𝑦 < 𝑧𝜓) ↔ (𝑦 < (𝑤 + 1) → 𝜓)))
98ralbidv 2380 . . 3 (𝑧 = (𝑤 + 1) → (∀𝑦 ∈ ℕ (𝑦 < 𝑧𝜓) ↔ ∀𝑦 ∈ ℕ (𝑦 < (𝑤 + 1) → 𝜓)))
10 breq2 3841 . . . . 5 (𝑧 = 𝑥 → (𝑦 < 𝑧𝑦 < 𝑥))
1110imbi1d 229 . . . 4 (𝑧 = 𝑥 → ((𝑦 < 𝑧𝜓) ↔ (𝑦 < 𝑥𝜓)))
1211ralbidv 2380 . . 3 (𝑧 = 𝑥 → (∀𝑦 ∈ ℕ (𝑦 < 𝑧𝜓) ↔ ∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓)))
13 nnnlt1 8420 . . . . 5 (𝑦 ∈ ℕ → ¬ 𝑦 < 1)
1413pm2.21d 584 . . . 4 (𝑦 ∈ ℕ → (𝑦 < 1 → 𝜓))
1514rgen 2428 . . 3 𝑦 ∈ ℕ (𝑦 < 1 → 𝜓)
16 1nn 8405 . . . . 5 1 ∈ ℕ
17 elex2 2635 . . . . 5 (1 ∈ ℕ → ∃𝑢 𝑢 ∈ ℕ)
18 nfra1 2409 . . . . . 6 𝑦𝑦 ∈ ℕ (𝑦 < 𝑤𝜓)
1918r19.3rm 3366 . . . . 5 (∃𝑢 𝑢 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) ↔ ∀𝑦 ∈ ℕ ∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓)))
2016, 17, 19mp2b 8 . . . 4 (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) ↔ ∀𝑦 ∈ ℕ ∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓))
21 rsp 2423 . . . . . . . . . 10 (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → (𝑦 ∈ ℕ → (𝑦 < 𝑤𝜓)))
2221com12 30 . . . . . . . . 9 (𝑦 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → (𝑦 < 𝑤𝜓)))
2322adantl 271 . . . . . . . 8 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → (𝑦 < 𝑤𝜓)))
24 indstr.2 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) → 𝜑))
2524rgen 2428 . . . . . . . . . . . 12 𝑥 ∈ ℕ (∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) → 𝜑)
26 nfv 1466 . . . . . . . . . . . . 13 𝑤(∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) → 𝜑)
27 nfv 1466 . . . . . . . . . . . . . 14 𝑥𝑦 ∈ ℕ (𝑦 < 𝑤𝜓)
28 nfsbc1v 2856 . . . . . . . . . . . . . 14 𝑥[𝑤 / 𝑥]𝜑
2927, 28nfim 1509 . . . . . . . . . . . . 13 𝑥(∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → [𝑤 / 𝑥]𝜑)
30 breq2 3841 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑤 → (𝑦 < 𝑥𝑦 < 𝑤))
3130imbi1d 229 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → ((𝑦 < 𝑥𝜓) ↔ (𝑦 < 𝑤𝜓)))
3231ralbidv 2380 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) ↔ ∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓)))
33 sbceq1a 2847 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝜑[𝑤 / 𝑥]𝜑))
3432, 33imbi12d 232 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → ((∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) → 𝜑) ↔ (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → [𝑤 / 𝑥]𝜑)))
3526, 29, 34cbvral 2586 . . . . . . . . . . . 12 (∀𝑥 ∈ ℕ (∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) → 𝜑) ↔ ∀𝑤 ∈ ℕ (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → [𝑤 / 𝑥]𝜑))
3625, 35mpbi 143 . . . . . . . . . . 11 𝑤 ∈ ℕ (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → [𝑤 / 𝑥]𝜑)
3736rspec 2427 . . . . . . . . . 10 (𝑤 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → [𝑤 / 𝑥]𝜑))
38 vex 2622 . . . . . . . . . . . . 13 𝑦 ∈ V
39 indstr.1 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝜑𝜓))
4038, 39sbcie 2871 . . . . . . . . . . . 12 ([𝑦 / 𝑥]𝜑𝜓)
41 dfsbcq 2840 . . . . . . . . . . . 12 (𝑦 = 𝑤 → ([𝑦 / 𝑥]𝜑[𝑤 / 𝑥]𝜑))
4240, 41syl5bbr 192 . . . . . . . . . . 11 (𝑦 = 𝑤 → (𝜓[𝑤 / 𝑥]𝜑))
4342biimprcd 158 . . . . . . . . . 10 ([𝑤 / 𝑥]𝜑 → (𝑦 = 𝑤𝜓))
4437, 43syl6 33 . . . . . . . . 9 (𝑤 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → (𝑦 = 𝑤𝜓)))
4544adantr 270 . . . . . . . 8 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → (𝑦 = 𝑤𝜓)))
4623, 45jcad 301 . . . . . . 7 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → ((𝑦 < 𝑤𝜓) ∧ (𝑦 = 𝑤𝜓))))
47 jaob 666 . . . . . . 7 (((𝑦 < 𝑤𝑦 = 𝑤) → 𝜓) ↔ ((𝑦 < 𝑤𝜓) ∧ (𝑦 = 𝑤𝜓)))
4846, 47syl6ibr 160 . . . . . 6 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → ((𝑦 < 𝑤𝑦 = 𝑤) → 𝜓)))
49 nnleltp1 8779 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (𝑦𝑤𝑦 < (𝑤 + 1)))
50 nnz 8739 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
51 nnz 8739 . . . . . . . . . 10 (𝑤 ∈ ℕ → 𝑤 ∈ ℤ)
52 zleloe 8767 . . . . . . . . . 10 ((𝑦 ∈ ℤ ∧ 𝑤 ∈ ℤ) → (𝑦𝑤 ↔ (𝑦 < 𝑤𝑦 = 𝑤)))
5350, 51, 52syl2an 283 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (𝑦𝑤 ↔ (𝑦 < 𝑤𝑦 = 𝑤)))
5449, 53bitr3d 188 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (𝑦 < (𝑤 + 1) ↔ (𝑦 < 𝑤𝑦 = 𝑤)))
5554ancoms 264 . . . . . . 7 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦 < (𝑤 + 1) ↔ (𝑦 < 𝑤𝑦 = 𝑤)))
5655imbi1d 229 . . . . . 6 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑦 < (𝑤 + 1) → 𝜓) ↔ ((𝑦 < 𝑤𝑦 = 𝑤) → 𝜓)))
5748, 56sylibrd 167 . . . . 5 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → (𝑦 < (𝑤 + 1) → 𝜓)))
5857ralimdva 2441 . . . 4 (𝑤 ∈ ℕ → (∀𝑦 ∈ ℕ ∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → ∀𝑦 ∈ ℕ (𝑦 < (𝑤 + 1) → 𝜓)))
5920, 58syl5bi 150 . . 3 (𝑤 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → ∀𝑦 ∈ ℕ (𝑦 < (𝑤 + 1) → 𝜓)))
603, 6, 9, 12, 15, 59nnind 8410 . 2 (𝑥 ∈ ℕ → ∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓))
6160, 24mpd 13 1 (𝑥 ∈ ℕ → 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 664   = wceq 1289  wex 1426  wcel 1438  wral 2359  [wsbc 2838   class class class wbr 3837  (class class class)co 5634  1c1 7330   + caddc 7332   < clt 7501  cle 7502  cn 8394  cz 8720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721
This theorem is referenced by:  indstr2  9065
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