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Theorem nnindf 32518
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 12222 . . . . . 6 1 ∈ ℕ
2 nnindf.5 . . . . . 6 𝜓
3 nnindf.1 . . . . . . 7 (𝑥 = 1 → (𝜑𝜓))
43elrab 3676 . . . . . 6 (1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (1 ∈ ℕ ∧ 𝜓))
51, 2, 4mpbir2an 708 . . . . 5 1 ∈ {𝑥 ∈ ℕ ∣ 𝜑}
6 elrabi 3670 . . . . . . . 8 (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑦 ∈ ℕ)
7 peano2nn 12223 . . . . . . . . . . 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 608 . . . . . . . . 9 (𝑦 ∈ ℕ → ((𝑦 ∈ ℕ ∧ 𝜒) → ((𝑦 + 1) ∈ ℕ ∧ 𝜃)))
11 nnindf.2 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝜑𝜒))
1211elrab 3676 . . . . . . . . 9 (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑦 ∈ ℕ ∧ 𝜒))
13 nnindf.3 . . . . . . . . . 10 (𝑥 = (𝑦 + 1) → (𝜑𝜃))
1413elrab 3676 . . . . . . . . 9 ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ((𝑦 + 1) ∈ ℕ ∧ 𝜃))
1510, 12, 143imtr4g 296 . . . . . . . 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 3055 . . . . . 6 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
18 nnindf.x . . . . . . . 8 𝑦𝜑
19 nfcv 2895 . . . . . . . 8 𝑦
2018, 19nfrabw 3460 . . . . . . 7 𝑦{𝑥 ∈ ℕ ∣ 𝜑}
21 nfcv 2895 . . . . . . 7 𝑤{𝑥 ∈ ℕ ∣ 𝜑}
22 nfv 1909 . . . . . . 7 𝑤(𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
2320nfel2 2913 . . . . . . 7 𝑦(𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
24 oveq1 7409 . . . . . . . 8 (𝑦 = 𝑤 → (𝑦 + 1) = (𝑤 + 1))
2524eleq1d 2810 . . . . . . 7 (𝑦 = 𝑤 → ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}))
2620, 21, 22, 23, 25cbvralfw 3293 . . . . . 6 (∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑})
2717, 26mpbi 229 . . . . 5 𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
28 peano5nni 12214 . . . . 5 ((1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}) → ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑})
295, 27, 28mp2an 689 . . . 4 ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑}
3029sseli 3971 . . 3 (𝐴 ∈ ℕ → 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑})
31 nnindf.4 . . . 4 (𝑥 = 𝐴 → (𝜑𝜏))
3231elrab 3676 . . 3 (𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝐴 ∈ ℕ ∧ 𝜏))
3330, 32sylib 217 . 2 (𝐴 ∈ ℕ → (𝐴 ∈ ℕ ∧ 𝜏))
3433simprd 495 1 (𝐴 ∈ ℕ → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wnf 1777  wcel 2098  wral 3053  {crab 3424  wss 3941  (class class class)co 7402  1c1 11108   + caddc 11110  cn 12211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719  ax-1cn 11165
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-om 7850  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-nn 12212
This theorem is referenced by:  nn0min  32519
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