Theorem nnindf 33507

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	⊢ Ⅎ𝑦𝜑
nnindf.1	⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓))
nnindf.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
nnindf.3	⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))
nnindf.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
nnindf.5	⊢ 𝜓
nnindf.6	⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃))

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.

Step	Hyp	Ref	Expression
1		1nn 12386	. . . . . 6 ⊢ 1 ∈ ℕ
2		nnindf.5	. . . . . 6 ⊢ 𝜓
3		nnindf.1	. . . . . . 7 ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓))
4	3	elrab 3648	. . . . . 6 ⊢ (1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (1 ∈ ℕ ∧ 𝜓))
5	1, 2, 4	mpbir2an 724	. . . . 5 ⊢ 1 ∈ {𝑥 ∈ ℕ ∣ 𝜑}
6		elrabi 3644	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑦 ∈ ℕ)
7		peano2nn 12387	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
8	7	a1d 26	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ))
9		nnindf.6	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃))
10	8, 9	anim12d 621	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → ((𝑦 ∈ ℕ ∧ 𝜒) → ((𝑦 + 1) ∈ ℕ ∧ 𝜃)))
11		nnindf.2	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
12	11	elrab 3648	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑦 ∈ ℕ ∧ 𝜒))
13		nnindf.3	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))
14	13	elrab 3648	. . . . . . . . 9 ⊢ ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ((𝑦 + 1) ∈ ℕ ∧ 𝜃))
15	10, 12, 14	3imtr4g 299	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ → (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}))
16	6, 15	mpcom 39	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑})
17	16	rgen 3082	. . . . . 6 ⊢ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
18		nnindf.x	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
19		nfcv 2926	. . . . . . . 8 ⊢ Ⅎ𝑦ℕ
20	18, 19	nfrabw 3451	. . . . . . 7 ⊢ Ⅎ𝑦{𝑥 ∈ ℕ ∣ 𝜑}
21		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑤{𝑥 ∈ ℕ ∣ 𝜑}
22		nfv 1949	. . . . . . 7 ⊢ Ⅎ𝑤(𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
23	20	nfel2 2944	. . . . . . 7 ⊢ Ⅎ𝑦(𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
24		oveq1 7458	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑦 + 1) = (𝑤 + 1))
25	24	eleq1d 2849	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}))
26	20, 21, 22, 23, 25	cbvralfw 3306	. . . . . 6 ⊢ (∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑})
27	17, 26	mpbi 233	. . . . 5 ⊢ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
28		peano5nni 12378	. . . . 5 ⊢ ((1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}) → ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑})
29	5, 27, 28	mp2an 705	. . . 4 ⊢ ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑}
30	29	sseli 3909	. . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑})
31		nnindf.4	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
32	31	elrab 3648	. . 3 ⊢ (𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝐴 ∈ ℕ ∧ 𝜏))
33	30, 32	sylib 221	. 2 ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ ℕ ∧ 𝜏))
34	33	simprd 501	1 ⊢ (𝐴 ∈ ℕ → 𝜏)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2147  ∀wral 3080  {crab 3416   ⊆ wss 3900  (class class class)co 7451  1c1 11240   + caddc 11242  ℕcn 12375

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783  ax-1cn 11297

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-nn 12376

This theorem is used by:  nn0min  33508