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Theorem nnindf 30129
 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 11387 . . . . . 6 1 ∈ ℕ
2 nnindf.5 . . . . . 6 𝜓
3 nnindf.1 . . . . . . 7 (𝑥 = 1 → (𝜑𝜓))
43elrab 3572 . . . . . 6 (1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (1 ∈ ℕ ∧ 𝜓))
51, 2, 4mpbir2an 701 . . . . 5 1 ∈ {𝑥 ∈ ℕ ∣ 𝜑}
6 elrabi 3567 . . . . . . . 8 (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑦 ∈ ℕ)
7 peano2nn 11388 . . . . . . . . . . 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 602 . . . . . . . . 9 (𝑦 ∈ ℕ → ((𝑦 ∈ ℕ ∧ 𝜒) → ((𝑦 + 1) ∈ ℕ ∧ 𝜃)))
11 nnindf.2 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝜑𝜒))
1211elrab 3572 . . . . . . . . 9 (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑦 ∈ ℕ ∧ 𝜒))
13 nnindf.3 . . . . . . . . . 10 (𝑥 = (𝑦 + 1) → (𝜑𝜃))
1413elrab 3572 . . . . . . . . 9 ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ((𝑦 + 1) ∈ ℕ ∧ 𝜃))
1510, 12, 143imtr4g 288 . . . . . . . 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 3104 . . . . . 6 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
18 nnindf.x . . . . . . . 8 𝑦𝜑
19 nfcv 2934 . . . . . . . 8 𝑦
2018, 19nfrab 3310 . . . . . . 7 𝑦{𝑥 ∈ ℕ ∣ 𝜑}
21 nfcv 2934 . . . . . . 7 𝑤{𝑥 ∈ ℕ ∣ 𝜑}
22 nfv 1957 . . . . . . 7 𝑤(𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
2320nfel2 2950 . . . . . . 7 𝑦(𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
24 oveq1 6929 . . . . . . . 8 (𝑦 = 𝑤 → (𝑦 + 1) = (𝑤 + 1))
2524eleq1d 2844 . . . . . . 7 (𝑦 = 𝑤 → ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}))
2620, 21, 22, 23, 25cbvralf 3361 . . . . . 6 (∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑})
2717, 26mpbi 222 . . . . 5 𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
28 peano5nni 11377 . . . . 5 ((1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}) → ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑})
295, 27, 28mp2an 682 . . . 4 ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑}
3029sseli 3817 . . 3 (𝐴 ∈ ℕ → 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑})
31 nnindf.4 . . . 4 (𝑥 = 𝐴 → (𝜑𝜏))
3231elrab 3572 . . 3 (𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝐴 ∈ ℕ ∧ 𝜏))
3330, 32sylib 210 . 2 (𝐴 ∈ ℕ → (𝐴 ∈ ℕ ∧ 𝜏))
3433simprd 491 1 (𝐴 ∈ ℕ → 𝜏)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601  Ⅎwnf 1827   ∈ wcel 2107  ∀wral 3090  {crab 3094   ⊆ wss 3792  (class class class)co 6922  1c1 10273   + caddc 10275  ℕcn 11374 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-1cn 10330 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-om 7344  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-nn 11375 This theorem is referenced by:  nn0min  30131
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