Theorem nnindf 32910

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 12168	. . . . . 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 712	. . . . 5 ⊢ 1 ∈ {𝑥 ∈ ℕ ∣ 𝜑}
6		elrabi 3644	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} → 𝑦 ∈ ℕ)
7		peano2nn 12169	. . . . . . . . . . 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 610	. . . . . . . . 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 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 3054	. . . . . 6 ⊢ ∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
18		nnindf.x	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
19		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑦ℕ
20	18, 19	nfrabw 3438	. . . . . . 7 ⊢ Ⅎ𝑦{𝑥 ∈ ℕ ∣ 𝜑}
21		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑤{𝑥 ∈ ℕ ∣ 𝜑}
22		nfv 1916	. . . . . . 7 ⊢ Ⅎ𝑤(𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
23	20	nfel2 2918	. . . . . . 7 ⊢ Ⅎ𝑦(𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}
24		oveq1 7375	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑦 + 1) = (𝑤 + 1))
25	24	eleq1d 2822	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ((𝑦 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑} ↔ (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}))
26	20, 21, 22, 23, 25	cbvralfw 3278	. . . . . 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 12160	. . . . 5 ⊢ ((1 ∈ {𝑥 ∈ ℕ ∣ 𝜑} ∧ ∀𝑤 ∈ {𝑥 ∈ ℕ ∣ 𝜑} (𝑤 + 1) ∈ {𝑥 ∈ ℕ ∣ 𝜑}) → ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑})
29	5, 27, 28	mp2an 693	. . . 4 ⊢ ℕ ⊆ {𝑥 ∈ ℕ ∣ 𝜑}
30	29	sseli 3931	. . 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 218	. 2 ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ ℕ ∧ 𝜏))
34	33	simprd 495	1 ⊢ (𝐴 ∈ ℕ → 𝜏)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114  ∀wral 3052  {crab 3401   ⊆ wss 3903  (class class class)co 7368  1c1 11039   + caddc 11041  ℕcn 12157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690  ax-1cn 11096

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-nn 12158

This theorem is referenced by:  nn0min  32911