Theorem nnindf 32826

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 12275	. . . . . 6 ⊢ 1 ∈ ℕ
2		nnindf.5	. . . . . 6 ⊢ 𝜓
3		nnindf.1	. . . . . . 7 ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓))
4	3	elrab 3695	. . . . . 6 ⊢ (1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (1 ∈ ℕ ∧ 𝜓))
5	1, 2, 4	mpbir2an 711	. . . . 5 ⊢ 1 ∈ {𝑥 ∈ ℕ ∣ 𝜑}
6		elrabi 3690	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑦 ∈ ℕ)
7		peano2nn 12276	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
8	7	a1d 25	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ))
9		nnindf.6	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃))
10	8, 9	anim12d 609	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → ((𝑦 ∈ ℕ ∧ 𝜒) → ((𝑦 + 1) ∈ ℕ ∧ 𝜃)))
11		nnindf.2	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
12	11	elrab 3695	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑦 ∈ ℕ ∧ 𝜒))
13		nnindf.3	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))
14	13	elrab 3695	. . . . . . . . 9 ⊢ ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ((𝑦 + 1) ∈ ℕ ∧ 𝜃))
15	10, 12, 14	3imtr4g 296	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ → (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}))
16	6, 15	mpcom 38	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑})
17	16	rgen 3061	. . . . . 6 ⊢ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
18		nnindf.x	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
19		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑦ℕ
20	18, 19	nfrabw 3473	. . . . . . 7 ⊢ Ⅎ𝑦{𝑥 ∈ ℕ ∣ 𝜑}
21		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑤{𝑥 ∈ ℕ ∣ 𝜑}
22		nfv 1912	. . . . . . 7 ⊢ Ⅎ𝑤(𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
23	20	nfel2 2922	. . . . . . 7 ⊢ Ⅎ𝑦(𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
24		oveq1 7438	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑦 + 1) = (𝑤 + 1))
25	24	eleq1d 2824	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}))
26	20, 21, 22, 23, 25	cbvralfw 3302	. . . . . 6 ⊢ (∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑})
27	17, 26	mpbi 230	. . . . 5 ⊢ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
28		peano5nni 12267	. . . . 5 ⊢ ((1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}) → ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑})
29	5, 27, 28	mp2an 692	. . . 4 ⊢ ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑}
30	29	sseli 3991	. . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑})
31		nnindf.4	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
32	31	elrab 3695	. . 3 ⊢ (𝐴 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝐴 ∈ ℕ ∧ 𝜏))
33	30, 32	sylib 218	. 2 ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ ℕ ∧ 𝜏))
34	33	simprd 495	1 ⊢ (𝐴 ∈ ℕ → 𝜏)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  Ⅎwnf 1780   ∈ wcel 2106  ∀wral 3059  {crab 3433   ⊆ wss 3963  (class class class)co 7431  1c1 11154   + caddc 11156  ℕcn 12264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-1cn 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-nn 12265

This theorem is referenced by:  nn0min  32827