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Theorem indstr 9592
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 4007 . . . . 5 (𝑧 = 1 → (𝑦 < 𝑧𝑦 < 1))
21imbi1d 231 . . . 4 (𝑧 = 1 → ((𝑦 < 𝑧𝜓) ↔ (𝑦 < 1 → 𝜓)))
32ralbidv 2477 . . 3 (𝑧 = 1 → (∀𝑦 ∈ ℕ (𝑦 < 𝑧𝜓) ↔ ∀𝑦 ∈ ℕ (𝑦 < 1 → 𝜓)))
4 breq2 4007 . . . . 5 (𝑧 = 𝑤 → (𝑦 < 𝑧𝑦 < 𝑤))
54imbi1d 231 . . . 4 (𝑧 = 𝑤 → ((𝑦 < 𝑧𝜓) ↔ (𝑦 < 𝑤𝜓)))
65ralbidv 2477 . . 3 (𝑧 = 𝑤 → (∀𝑦 ∈ ℕ (𝑦 < 𝑧𝜓) ↔ ∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓)))
7 breq2 4007 . . . . 5 (𝑧 = (𝑤 + 1) → (𝑦 < 𝑧𝑦 < (𝑤 + 1)))
87imbi1d 231 . . . 4 (𝑧 = (𝑤 + 1) → ((𝑦 < 𝑧𝜓) ↔ (𝑦 < (𝑤 + 1) → 𝜓)))
98ralbidv 2477 . . 3 (𝑧 = (𝑤 + 1) → (∀𝑦 ∈ ℕ (𝑦 < 𝑧𝜓) ↔ ∀𝑦 ∈ ℕ (𝑦 < (𝑤 + 1) → 𝜓)))
10 breq2 4007 . . . . 5 (𝑧 = 𝑥 → (𝑦 < 𝑧𝑦 < 𝑥))
1110imbi1d 231 . . . 4 (𝑧 = 𝑥 → ((𝑦 < 𝑧𝜓) ↔ (𝑦 < 𝑥𝜓)))
1211ralbidv 2477 . . 3 (𝑧 = 𝑥 → (∀𝑦 ∈ ℕ (𝑦 < 𝑧𝜓) ↔ ∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓)))
13 nnnlt1 8944 . . . . 5 (𝑦 ∈ ℕ → ¬ 𝑦 < 1)
1413pm2.21d 619 . . . 4 (𝑦 ∈ ℕ → (𝑦 < 1 → 𝜓))
1514rgen 2530 . . 3 𝑦 ∈ ℕ (𝑦 < 1 → 𝜓)
16 1nn 8929 . . . . 5 1 ∈ ℕ
17 elex2 2753 . . . . 5 (1 ∈ ℕ → ∃𝑢 𝑢 ∈ ℕ)
18 nfra1 2508 . . . . . 6 𝑦𝑦 ∈ ℕ (𝑦 < 𝑤𝜓)
1918r19.3rm 3511 . . . . 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 2524 . . . . . . . . . 10 (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → (𝑦 ∈ ℕ → (𝑦 < 𝑤𝜓)))
2221com12 30 . . . . . . . . 9 (𝑦 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → (𝑦 < 𝑤𝜓)))
2322adantl 277 . . . . . . . 8 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → (𝑦 < 𝑤𝜓)))
24 indstr.2 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) → 𝜑))
2524rgen 2530 . . . . . . . . . . . 12 𝑥 ∈ ℕ (∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) → 𝜑)
26 nfv 1528 . . . . . . . . . . . . 13 𝑤(∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) → 𝜑)
27 nfv 1528 . . . . . . . . . . . . . 14 𝑥𝑦 ∈ ℕ (𝑦 < 𝑤𝜓)
28 nfsbc1v 2981 . . . . . . . . . . . . . 14 𝑥[𝑤 / 𝑥]𝜑
2927, 28nfim 1572 . . . . . . . . . . . . 13 𝑥(∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → [𝑤 / 𝑥]𝜑)
30 breq2 4007 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑤 → (𝑦 < 𝑥𝑦 < 𝑤))
3130imbi1d 231 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → ((𝑦 < 𝑥𝜓) ↔ (𝑦 < 𝑤𝜓)))
3231ralbidv 2477 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) ↔ ∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓)))
33 sbceq1a 2972 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝜑[𝑤 / 𝑥]𝜑))
3432, 33imbi12d 234 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → ((∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) → 𝜑) ↔ (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → [𝑤 / 𝑥]𝜑)))
3526, 29, 34cbvral 2699 . . . . . . . . . . . 12 (∀𝑥 ∈ ℕ (∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) → 𝜑) ↔ ∀𝑤 ∈ ℕ (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → [𝑤 / 𝑥]𝜑))
3625, 35mpbi 145 . . . . . . . . . . 11 𝑤 ∈ ℕ (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → [𝑤 / 𝑥]𝜑)
3736rspec 2529 . . . . . . . . . 10 (𝑤 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → [𝑤 / 𝑥]𝜑))
38 vex 2740 . . . . . . . . . . . . 13 𝑦 ∈ V
39 indstr.1 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝜑𝜓))
4038, 39sbcie 2997 . . . . . . . . . . . 12 ([𝑦 / 𝑥]𝜑𝜓)
41 dfsbcq 2964 . . . . . . . . . . . 12 (𝑦 = 𝑤 → ([𝑦 / 𝑥]𝜑[𝑤 / 𝑥]𝜑))
4240, 41bitr3id 194 . . . . . . . . . . 11 (𝑦 = 𝑤 → (𝜓[𝑤 / 𝑥]𝜑))
4342biimprcd 160 . . . . . . . . . 10 ([𝑤 / 𝑥]𝜑 → (𝑦 = 𝑤𝜓))
4437, 43syl6 33 . . . . . . . . 9 (𝑤 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → (𝑦 = 𝑤𝜓)))
4544adantr 276 . . . . . . . 8 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → (𝑦 = 𝑤𝜓)))
4623, 45jcad 307 . . . . . . 7 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → ((𝑦 < 𝑤𝜓) ∧ (𝑦 = 𝑤𝜓))))
47 jaob 710 . . . . . . 7 (((𝑦 < 𝑤𝑦 = 𝑤) → 𝜓) ↔ ((𝑦 < 𝑤𝜓) ∧ (𝑦 = 𝑤𝜓)))
4846, 47imbitrrdi 162 . . . . . 6 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → ((𝑦 < 𝑤𝑦 = 𝑤) → 𝜓)))
49 nnleltp1 9311 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (𝑦𝑤𝑦 < (𝑤 + 1)))
50 nnz 9271 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
51 nnz 9271 . . . . . . . . . 10 (𝑤 ∈ ℕ → 𝑤 ∈ ℤ)
52 zleloe 9299 . . . . . . . . . 10 ((𝑦 ∈ ℤ ∧ 𝑤 ∈ ℤ) → (𝑦𝑤 ↔ (𝑦 < 𝑤𝑦 = 𝑤)))
5350, 51, 52syl2an 289 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (𝑦𝑤 ↔ (𝑦 < 𝑤𝑦 = 𝑤)))
5449, 53bitr3d 190 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (𝑦 < (𝑤 + 1) ↔ (𝑦 < 𝑤𝑦 = 𝑤)))
5554ancoms 268 . . . . . . 7 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦 < (𝑤 + 1) ↔ (𝑦 < 𝑤𝑦 = 𝑤)))
5655imbi1d 231 . . . . . 6 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑦 < (𝑤 + 1) → 𝜓) ↔ ((𝑦 < 𝑤𝑦 = 𝑤) → 𝜓)))
5748, 56sylibrd 169 . . . . 5 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → (𝑦 < (𝑤 + 1) → 𝜓)))
5857ralimdva 2544 . . . 4 (𝑤 ∈ ℕ → (∀𝑦 ∈ ℕ ∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → ∀𝑦 ∈ ℕ (𝑦 < (𝑤 + 1) → 𝜓)))
5920, 58biimtrid 152 . . 3 (𝑤 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → ∀𝑦 ∈ ℕ (𝑦 < (𝑤 + 1) → 𝜓)))
603, 6, 9, 12, 15, 59nnind 8934 . 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 104  wb 105  wo 708   = wceq 1353  wex 1492  wcel 2148  wral 2455  [wsbc 2962   class class class wbr 4003  (class class class)co 5874  1c1 7811   + caddc 7813   < clt 7991  cle 7992  cn 8918  cz 9252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-inn 8919  df-n0 9176  df-z 9253
This theorem is referenced by:  indstr2  9608
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