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Theorem indstr 9483
 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 3965 . . . . 5 (𝑧 = 1 → (𝑦 < 𝑧𝑦 < 1))
21imbi1d 230 . . . 4 (𝑧 = 1 → ((𝑦 < 𝑧𝜓) ↔ (𝑦 < 1 → 𝜓)))
32ralbidv 2454 . . 3 (𝑧 = 1 → (∀𝑦 ∈ ℕ (𝑦 < 𝑧𝜓) ↔ ∀𝑦 ∈ ℕ (𝑦 < 1 → 𝜓)))
4 breq2 3965 . . . . 5 (𝑧 = 𝑤 → (𝑦 < 𝑧𝑦 < 𝑤))
54imbi1d 230 . . . 4 (𝑧 = 𝑤 → ((𝑦 < 𝑧𝜓) ↔ (𝑦 < 𝑤𝜓)))
65ralbidv 2454 . . 3 (𝑧 = 𝑤 → (∀𝑦 ∈ ℕ (𝑦 < 𝑧𝜓) ↔ ∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓)))
7 breq2 3965 . . . . 5 (𝑧 = (𝑤 + 1) → (𝑦 < 𝑧𝑦 < (𝑤 + 1)))
87imbi1d 230 . . . 4 (𝑧 = (𝑤 + 1) → ((𝑦 < 𝑧𝜓) ↔ (𝑦 < (𝑤 + 1) → 𝜓)))
98ralbidv 2454 . . 3 (𝑧 = (𝑤 + 1) → (∀𝑦 ∈ ℕ (𝑦 < 𝑧𝜓) ↔ ∀𝑦 ∈ ℕ (𝑦 < (𝑤 + 1) → 𝜓)))
10 breq2 3965 . . . . 5 (𝑧 = 𝑥 → (𝑦 < 𝑧𝑦 < 𝑥))
1110imbi1d 230 . . . 4 (𝑧 = 𝑥 → ((𝑦 < 𝑧𝜓) ↔ (𝑦 < 𝑥𝜓)))
1211ralbidv 2454 . . 3 (𝑧 = 𝑥 → (∀𝑦 ∈ ℕ (𝑦 < 𝑧𝜓) ↔ ∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓)))
13 nnnlt1 8838 . . . . 5 (𝑦 ∈ ℕ → ¬ 𝑦 < 1)
1413pm2.21d 609 . . . 4 (𝑦 ∈ ℕ → (𝑦 < 1 → 𝜓))
1514rgen 2507 . . 3 𝑦 ∈ ℕ (𝑦 < 1 → 𝜓)
16 1nn 8823 . . . . 5 1 ∈ ℕ
17 elex2 2725 . . . . 5 (1 ∈ ℕ → ∃𝑢 𝑢 ∈ ℕ)
18 nfra1 2485 . . . . . 6 𝑦𝑦 ∈ ℕ (𝑦 < 𝑤𝜓)
1918r19.3rm 3478 . . . . 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 2501 . . . . . . . . . 10 (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → (𝑦 ∈ ℕ → (𝑦 < 𝑤𝜓)))
2221com12 30 . . . . . . . . 9 (𝑦 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → (𝑦 < 𝑤𝜓)))
2322adantl 275 . . . . . . . 8 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → (𝑦 < 𝑤𝜓)))
24 indstr.2 . . . . . . . . . . . . 13 (𝑥 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) → 𝜑))
2524rgen 2507 . . . . . . . . . . . 12 𝑥 ∈ ℕ (∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) → 𝜑)
26 nfv 1505 . . . . . . . . . . . . 13 𝑤(∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) → 𝜑)
27 nfv 1505 . . . . . . . . . . . . . 14 𝑥𝑦 ∈ ℕ (𝑦 < 𝑤𝜓)
28 nfsbc1v 2951 . . . . . . . . . . . . . 14 𝑥[𝑤 / 𝑥]𝜑
2927, 28nfim 1549 . . . . . . . . . . . . 13 𝑥(∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → [𝑤 / 𝑥]𝜑)
30 breq2 3965 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑤 → (𝑦 < 𝑥𝑦 < 𝑤))
3130imbi1d 230 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → ((𝑦 < 𝑥𝜓) ↔ (𝑦 < 𝑤𝜓)))
3231ralbidv 2454 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) ↔ ∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓)))
33 sbceq1a 2942 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝜑[𝑤 / 𝑥]𝜑))
3432, 33imbi12d 233 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → ((∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) → 𝜑) ↔ (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → [𝑤 / 𝑥]𝜑)))
3526, 29, 34cbvral 2673 . . . . . . . . . . . 12 (∀𝑥 ∈ ℕ (∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) → 𝜑) ↔ ∀𝑤 ∈ ℕ (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → [𝑤 / 𝑥]𝜑))
3625, 35mpbi 144 . . . . . . . . . . 11 𝑤 ∈ ℕ (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → [𝑤 / 𝑥]𝜑)
3736rspec 2506 . . . . . . . . . 10 (𝑤 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → [𝑤 / 𝑥]𝜑))
38 vex 2712 . . . . . . . . . . . . 13 𝑦 ∈ V
39 indstr.1 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝜑𝜓))
4038, 39sbcie 2967 . . . . . . . . . . . 12 ([𝑦 / 𝑥]𝜑𝜓)
41 dfsbcq 2935 . . . . . . . . . . . 12 (𝑦 = 𝑤 → ([𝑦 / 𝑥]𝜑[𝑤 / 𝑥]𝜑))
4240, 41bitr3id 193 . . . . . . . . . . 11 (𝑦 = 𝑤 → (𝜓[𝑤 / 𝑥]𝜑))
4342biimprcd 159 . . . . . . . . . 10 ([𝑤 / 𝑥]𝜑 → (𝑦 = 𝑤𝜓))
4437, 43syl6 33 . . . . . . . . 9 (𝑤 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → (𝑦 = 𝑤𝜓)))
4544adantr 274 . . . . . . . 8 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → (𝑦 = 𝑤𝜓)))
4623, 45jcad 305 . . . . . . 7 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → ((𝑦 < 𝑤𝜓) ∧ (𝑦 = 𝑤𝜓))))
47 jaob 700 . . . . . . 7 (((𝑦 < 𝑤𝑦 = 𝑤) → 𝜓) ↔ ((𝑦 < 𝑤𝜓) ∧ (𝑦 = 𝑤𝜓)))
4846, 47syl6ibr 161 . . . . . 6 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → ((𝑦 < 𝑤𝑦 = 𝑤) → 𝜓)))
49 nnleltp1 9205 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (𝑦𝑤𝑦 < (𝑤 + 1)))
50 nnz 9165 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
51 nnz 9165 . . . . . . . . . 10 (𝑤 ∈ ℕ → 𝑤 ∈ ℤ)
52 zleloe 9193 . . . . . . . . . 10 ((𝑦 ∈ ℤ ∧ 𝑤 ∈ ℤ) → (𝑦𝑤 ↔ (𝑦 < 𝑤𝑦 = 𝑤)))
5350, 51, 52syl2an 287 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (𝑦𝑤 ↔ (𝑦 < 𝑤𝑦 = 𝑤)))
5449, 53bitr3d 189 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (𝑦 < (𝑤 + 1) ↔ (𝑦 < 𝑤𝑦 = 𝑤)))
5554ancoms 266 . . . . . . 7 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦 < (𝑤 + 1) ↔ (𝑦 < 𝑤𝑦 = 𝑤)))
5655imbi1d 230 . . . . . 6 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑦 < (𝑤 + 1) → 𝜓) ↔ ((𝑦 < 𝑤𝑦 = 𝑤) → 𝜓)))
5748, 56sylibrd 168 . . . . 5 ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → (𝑦 < (𝑤 + 1) → 𝜓)))
5857ralimdva 2521 . . . 4 (𝑤 ∈ ℕ → (∀𝑦 ∈ ℕ ∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → ∀𝑦 ∈ ℕ (𝑦 < (𝑤 + 1) → 𝜓)))
5920, 58syl5bi 151 . . 3 (𝑤 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑤𝜓) → ∀𝑦 ∈ ℕ (𝑦 < (𝑤 + 1) → 𝜓)))
603, 6, 9, 12, 15, 59nnind 8828 . 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 103   ↔ wb 104   ∨ wo 698   = wceq 1332  ∃wex 1469   ∈ wcel 2125  ∀wral 2432  [wsbc 2933   class class class wbr 3961  (class class class)co 5814  1c1 7712   + caddc 7714   < clt 7891   ≤ cle 7892  ℕcn 8812  ℤcz 9146 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-cnex 7802  ax-resscn 7803  ax-1cn 7804  ax-1re 7805  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-addcom 7811  ax-addass 7813  ax-distr 7815  ax-i2m1 7816  ax-0lt1 7817  ax-0id 7819  ax-rnegex 7820  ax-cnre 7822  ax-pre-ltirr 7823  ax-pre-ltwlin 7824  ax-pre-lttrn 7825  ax-pre-ltadd 7827 This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-reu 2439  df-rab 2441  df-v 2711  df-sbc 2934  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-br 3962  df-opab 4022  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-iota 5128  df-fun 5165  df-fv 5171  df-riota 5770  df-ov 5817  df-oprab 5818  df-mpo 5819  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896  df-le 7897  df-sub 8027  df-neg 8028  df-inn 8813  df-n0 9070  df-z 9147 This theorem is referenced by:  indstr2  9498
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