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Theorem nnindf 28748
Description: Principle of Mathematical Induction, using a bound-variable hypothesis instead of distinct variables. (Contributed by Thierry Arnoux, 6-May-2018.)
Hypotheses
Ref Expression
nnindf.x 𝑦𝜑
nnindf.1 (𝑥 = 1 → (𝜑𝜓))
nnindf.2 (𝑥 = 𝑦 → (𝜑𝜒))
nnindf.3 (𝑥 = (𝑦 + 1) → (𝜑𝜃))
nnindf.4 (𝑥 = 𝐴 → (𝜑𝜏))
nnindf.5 𝜓
nnindf.6 (𝑦 ∈ ℕ → (𝜒𝜃))
Assertion
Ref Expression
nnindf (𝐴 ∈ ℕ → 𝜏)
Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝜒,𝑥   𝜓,𝑥   𝜏,𝑥   𝜃,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑦)   𝜃(𝑦)   𝜏(𝑦)   𝐴(𝑦)

Proof of Theorem nnindf
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 1nn 10785 . . . . . 6 1 ∈ ℕ
2 nnindf.5 . . . . . 6 𝜓
3 nnindf.1 . . . . . . 7 (𝑥 = 1 → (𝜑𝜓))
43elrab 3235 . . . . . 6 (1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (1 ∈ ℕ ∧ 𝜓))
51, 2, 4mpbir2an 956 . . . . 5 1 ∈ {𝑥 ∈ ℕ ∣ 𝜑}
6 elrabi 3232 . . . . . . . 8 (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑦 ∈ ℕ)
7 peano2nn 10786 . . . . . . . . . . 11 (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
87a1d 25 . . . . . . . . . 10 (𝑦 ∈ ℕ → (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ))
9 nnindf.6 . . . . . . . . . 10 (𝑦 ∈ ℕ → (𝜒𝜃))
108, 9anim12d 583 . . . . . . . . 9 (𝑦 ∈ ℕ → ((𝑦 ∈ ℕ ∧ 𝜒) → ((𝑦 + 1) ∈ ℕ ∧ 𝜃)))
11 nnindf.2 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝜑𝜒))
1211elrab 3235 . . . . . . . . 9 (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑦 ∈ ℕ ∧ 𝜒))
13 nnindf.3 . . . . . . . . . 10 (𝑥 = (𝑦 + 1) → (𝜑𝜃))
1413elrab 3235 . . . . . . . . 9 ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ((𝑦 + 1) ∈ ℕ ∧ 𝜃))
1510, 12, 143imtr4g 283 . . . . . . . 8 (𝑦 ∈ ℕ → (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}))
166, 15mpcom 37 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑})
1716rgen 2810 . . . . . 6 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
18 nnindf.x . . . . . . . 8 𝑦𝜑
19 nfcv 2655 . . . . . . . 8 𝑦
2018, 19nfrab 3004 . . . . . . 7 𝑦{𝑥 ∈ ℕ ∣ 𝜑}
21 nfcv 2655 . . . . . . 7 𝑤{𝑥 ∈ ℕ ∣ 𝜑}
22 nfv 1796 . . . . . . 7 𝑤(𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
2320nfel2 2671 . . . . . . 7 𝑦(𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
24 oveq1 6432 . . . . . . . 8 (𝑦 = 𝑤 → (𝑦 + 1) = (𝑤 + 1))
2524eleq1d 2576 . . . . . . 7 (𝑦 = 𝑤 → ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}))
2620, 21, 22, 23, 25cbvralf 3045 . . . . . 6 (∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑})
2717, 26mpbi 218 . . . . 5 𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
28 peano5nni 10777 . . . . 5 ((1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}) → ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑})
295, 27, 28mp2an 703 . . . 4 ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑}
3029sseli 3468 . . 3 (𝐴 ∈ ℕ → 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑})
31 nnindf.4 . . . 4 (𝑥 = 𝐴 → (𝜑𝜏))
3231elrab 3235 . . 3 (𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝐴 ∈ ℕ ∧ 𝜏))
3330, 32sylib 206 . 2 (𝐴 ∈ ℕ → (𝐴 ∈ ℕ ∧ 𝜏))
3433simprd 477 1 (𝐴 ∈ ℕ → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wnf 1698  wcel 1938  wral 2800  {crab 2804  wss 3444  (class class class)co 6425  1c1 9691   + caddc 9693  cn 10774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6722  ax-1cn 9748
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6428  df-om 6833  df-wrecs 7168  df-recs 7230  df-rdg 7268  df-nn 10775
This theorem is referenced by:  nn0min  28750
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