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Theorem nnindf 29693
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 11069 . . . . . 6 1 ∈ ℕ
2 nnindf.5 . . . . . 6 𝜓
3 nnindf.1 . . . . . . 7 (𝑥 = 1 → (𝜑𝜓))
43elrab 3396 . . . . . 6 (1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (1 ∈ ℕ ∧ 𝜓))
51, 2, 4mpbir2an 975 . . . . 5 1 ∈ {𝑥 ∈ ℕ ∣ 𝜑}
6 elrabi 3391 . . . . . . . 8 (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑦 ∈ ℕ)
7 peano2nn 11070 . . . . . . . . . . 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 585 . . . . . . . . 9 (𝑦 ∈ ℕ → ((𝑦 ∈ ℕ ∧ 𝜒) → ((𝑦 + 1) ∈ ℕ ∧ 𝜃)))
11 nnindf.2 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝜑𝜒))
1211elrab 3396 . . . . . . . . 9 (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑦 ∈ ℕ ∧ 𝜒))
13 nnindf.3 . . . . . . . . . 10 (𝑥 = (𝑦 + 1) → (𝜑𝜃))
1413elrab 3396 . . . . . . . . 9 ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ((𝑦 + 1) ∈ ℕ ∧ 𝜃))
1510, 12, 143imtr4g 285 . . . . . . . 8 (𝑦 ∈ ℕ → (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}))
166, 15mpcom 38 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑})
1716rgen 2951 . . . . . 6 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
18 nnindf.x . . . . . . . 8 𝑦𝜑
19 nfcv 2793 . . . . . . . 8 𝑦
2018, 19nfrab 3153 . . . . . . 7 𝑦{𝑥 ∈ ℕ ∣ 𝜑}
21 nfcv 2793 . . . . . . 7 𝑤{𝑥 ∈ ℕ ∣ 𝜑}
22 nfv 1883 . . . . . . 7 𝑤(𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
2320nfel2 2810 . . . . . . 7 𝑦(𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
24 oveq1 6697 . . . . . . . 8 (𝑦 = 𝑤 → (𝑦 + 1) = (𝑤 + 1))
2524eleq1d 2715 . . . . . . 7 (𝑦 = 𝑤 → ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}))
2620, 21, 22, 23, 25cbvralf 3195 . . . . . 6 (∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑})
2717, 26mpbi 220 . . . . 5 𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
28 peano5nni 11061 . . . . 5 ((1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}) → ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑})
295, 27, 28mp2an 708 . . . 4 ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑}
3029sseli 3632 . . 3 (𝐴 ∈ ℕ → 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑})
31 nnindf.4 . . . 4 (𝑥 = 𝐴 → (𝜑𝜏))
3231elrab 3396 . . 3 (𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝐴 ∈ ℕ ∧ 𝜏))
3330, 32sylib 208 . 2 (𝐴 ∈ ℕ → (𝐴 ∈ ℕ ∧ 𝜏))
3433simprd 478 1 (𝐴 ∈ ℕ → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wnf 1748  wcel 2030  wral 2941  {crab 2945  wss 3607  (class class class)co 6690  1c1 9975   + caddc 9977  cn 11058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-1cn 10032
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-nn 11059
This theorem is referenced by:  nn0min  29695
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