Theorem indstr 9667

Description: Strong Mathematical Induction for positive integers (inference schema). (Contributed by NM, 17-Aug-2001.)

Hypotheses

Ref	Expression
indstr.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
indstr.2	⊢ (𝑥 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑥 → 𝜓) → 𝜑))

Assertion

Ref	Expression
indstr	⊢ (𝑥 ∈ ℕ → 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem indstr

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4037	. . . . 5 ⊢ (𝑧 = 1 → (𝑦 < 𝑧 ↔ 𝑦 < 1))
2	1	imbi1d 231	. . . 4 ⊢ (𝑧 = 1 → ((𝑦 < 𝑧 → 𝜓) ↔ (𝑦 < 1 → 𝜓)))
3	2	ralbidv 2497	. . 3 ⊢ (𝑧 = 1 → (∀𝑦 ∈ ℕ (𝑦 < 𝑧 → 𝜓) ↔ ∀𝑦 ∈ ℕ (𝑦 < 1 → 𝜓)))
4		breq2 4037	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑦 < 𝑧 ↔ 𝑦 < 𝑤))
5	4	imbi1d 231	. . . 4 ⊢ (𝑧 = 𝑤 → ((𝑦 < 𝑧 → 𝜓) ↔ (𝑦 < 𝑤 → 𝜓)))
6	5	ralbidv 2497	. . 3 ⊢ (𝑧 = 𝑤 → (∀𝑦 ∈ ℕ (𝑦 < 𝑧 → 𝜓) ↔ ∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓)))
7		breq2 4037	. . . . 5 ⊢ (𝑧 = (𝑤 + 1) → (𝑦 < 𝑧 ↔ 𝑦 < (𝑤 + 1)))
8	7	imbi1d 231	. . . 4 ⊢ (𝑧 = (𝑤 + 1) → ((𝑦 < 𝑧 → 𝜓) ↔ (𝑦 < (𝑤 + 1) → 𝜓)))
9	8	ralbidv 2497	. . 3 ⊢ (𝑧 = (𝑤 + 1) → (∀𝑦 ∈ ℕ (𝑦 < 𝑧 → 𝜓) ↔ ∀𝑦 ∈ ℕ (𝑦 < (𝑤 + 1) → 𝜓)))
10		breq2 4037	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝑦 < 𝑧 ↔ 𝑦 < 𝑥))
11	10	imbi1d 231	. . . 4 ⊢ (𝑧 = 𝑥 → ((𝑦 < 𝑧 → 𝜓) ↔ (𝑦 < 𝑥 → 𝜓)))
12	11	ralbidv 2497	. . 3 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ ℕ (𝑦 < 𝑧 → 𝜓) ↔ ∀𝑦 ∈ ℕ (𝑦 < 𝑥 → 𝜓)))
13		nnnlt1 9016	. . . . 5 ⊢ (𝑦 ∈ ℕ → ¬ 𝑦 < 1)
14	13	pm2.21d 620	. . . 4 ⊢ (𝑦 ∈ ℕ → (𝑦 < 1 → 𝜓))
15	14	rgen 2550	. . 3 ⊢ ∀𝑦 ∈ ℕ (𝑦 < 1 → 𝜓)
16		1nn 9001	. . . . 5 ⊢ 1 ∈ ℕ
17		elex2 2779	. . . . 5 ⊢ (1 ∈ ℕ → ∃𝑢 𝑢 ∈ ℕ)
18		nfra1 2528	. . . . . 6 ⊢ Ⅎ𝑦∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓)
19	18	r19.3rm 3539	. . . . 5 ⊢ (∃𝑢 𝑢 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓) ↔ ∀𝑦 ∈ ℕ ∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓)))
20	16, 17, 19	mp2b 8	. . . 4 ⊢ (∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓) ↔ ∀𝑦 ∈ ℕ ∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓))
21		rsp 2544	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓) → (𝑦 ∈ ℕ → (𝑦 < 𝑤 → 𝜓)))
22	21	com12 30	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓) → (𝑦 < 𝑤 → 𝜓)))
23	22	adantl 277	. . . . . . . 8 ⊢ ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓) → (𝑦 < 𝑤 → 𝜓)))
24		indstr.2	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑥 → 𝜓) → 𝜑))
25	24	rgen 2550	. . . . . . . . . . . 12 ⊢ ∀𝑥 ∈ ℕ (∀𝑦 ∈ ℕ (𝑦 < 𝑥 → 𝜓) → 𝜑)
26		nfv 1542	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤(∀𝑦 ∈ ℕ (𝑦 < 𝑥 → 𝜓) → 𝜑)
27		nfv 1542	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓)
28		nfsbc1v 3008	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑
29	27, 28	nfim 1586	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓) → [𝑤 / 𝑥]𝜑)
30		breq2 4037	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → (𝑦 < 𝑥 ↔ 𝑦 < 𝑤))
31	30	imbi1d 231	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → ((𝑦 < 𝑥 → 𝜓) ↔ (𝑦 < 𝑤 → 𝜓)))
32	31	ralbidv 2497	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ ℕ (𝑦 < 𝑥 → 𝜓) ↔ ∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓)))
33		sbceq1a 2999	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
34	32, 33	imbi12d 234	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → ((∀𝑦 ∈ ℕ (𝑦 < 𝑥 → 𝜓) → 𝜑) ↔ (∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓) → [𝑤 / 𝑥]𝜑)))
35	26, 29, 34	cbvral 2725	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ ℕ (∀𝑦 ∈ ℕ (𝑦 < 𝑥 → 𝜓) → 𝜑) ↔ ∀𝑤 ∈ ℕ (∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓) → [𝑤 / 𝑥]𝜑))
36	25, 35	mpbi 145	. . . . . . . . . . 11 ⊢ ∀𝑤 ∈ ℕ (∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓) → [𝑤 / 𝑥]𝜑)
37	36	rspec 2549	. . . . . . . . . 10 ⊢ (𝑤 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓) → [𝑤 / 𝑥]𝜑))
38		vex 2766	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
39		indstr.1	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
40	38, 39	sbcie 3024	. . . . . . . . . . . 12 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
41		dfsbcq 2991	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → ([𝑦 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑))
42	40, 41	bitr3id 194	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑥]𝜑))
43	42	biimprcd 160	. . . . . . . . . 10 ⊢ ([𝑤 / 𝑥]𝜑 → (𝑦 = 𝑤 → 𝜓))
44	37, 43	syl6 33	. . . . . . . . 9 ⊢ (𝑤 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓) → (𝑦 = 𝑤 → 𝜓)))
45	44	adantr 276	. . . . . . . 8 ⊢ ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓) → (𝑦 = 𝑤 → 𝜓)))
46	23, 45	jcad 307	. . . . . . 7 ⊢ ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓) → ((𝑦 < 𝑤 → 𝜓) ∧ (𝑦 = 𝑤 → 𝜓))))
47		jaob 711	. . . . . . 7 ⊢ (((𝑦 < 𝑤 ∨ 𝑦 = 𝑤) → 𝜓) ↔ ((𝑦 < 𝑤 → 𝜓) ∧ (𝑦 = 𝑤 → 𝜓)))
48	46, 47	imbitrrdi 162	. . . . . 6 ⊢ ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓) → ((𝑦 < 𝑤 ∨ 𝑦 = 𝑤) → 𝜓)))
49		nnleltp1 9385	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (𝑦 ≤ 𝑤 ↔ 𝑦 < (𝑤 + 1)))
50		nnz 9345	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
51		nnz 9345	. . . . . . . . . 10 ⊢ (𝑤 ∈ ℕ → 𝑤 ∈ ℤ)
52		zleloe 9373	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℤ ∧ 𝑤 ∈ ℤ) → (𝑦 ≤ 𝑤 ↔ (𝑦 < 𝑤 ∨ 𝑦 = 𝑤)))
53	50, 51, 52	syl2an 289	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (𝑦 ≤ 𝑤 ↔ (𝑦 < 𝑤 ∨ 𝑦 = 𝑤)))
54	49, 53	bitr3d 190	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (𝑦 < (𝑤 + 1) ↔ (𝑦 < 𝑤 ∨ 𝑦 = 𝑤)))
55	54	ancoms 268	. . . . . . 7 ⊢ ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦 < (𝑤 + 1) ↔ (𝑦 < 𝑤 ∨ 𝑦 = 𝑤)))
56	55	imbi1d 231	. . . . . 6 ⊢ ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑦 < (𝑤 + 1) → 𝜓) ↔ ((𝑦 < 𝑤 ∨ 𝑦 = 𝑤) → 𝜓)))
57	48, 56	sylibrd 169	. . . . 5 ⊢ ((𝑤 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓) → (𝑦 < (𝑤 + 1) → 𝜓)))
58	57	ralimdva 2564	. . . 4 ⊢ (𝑤 ∈ ℕ → (∀𝑦 ∈ ℕ ∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓) → ∀𝑦 ∈ ℕ (𝑦 < (𝑤 + 1) → 𝜓)))
59	20, 58	biimtrid 152	. . 3 ⊢ (𝑤 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑤 → 𝜓) → ∀𝑦 ∈ ℕ (𝑦 < (𝑤 + 1) → 𝜓)))
60	3, 6, 9, 12, 15, 59	nnind 9006	. 2 ⊢ (𝑥 ∈ ℕ → ∀𝑦 ∈ ℕ (𝑦 < 𝑥 → 𝜓))
61	60, 24	mpd 13	1 ⊢ (𝑥 ∈ ℕ → 𝜑)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∀wral 2475  [wsbc 2989   class class class wbr 4033  (class class class)co 5922  1c1 7880   + caddc 7882   < clt 8061   ≤ cle 8062  ℕcn 8990  ℤcz 9326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327

This theorem is referenced by:  indstr2  9683