Theorem bj-findis 10774

Description: Principle of induction, using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See bj-bdfindis 10742 for a bounded version not requiring ax-setind 4280. See finds 4341 for a proof in IZF. From this version, it is easy to prove of finds 4341, finds2 4342, finds1 4343. (Contributed by BJ, 22-Dec-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-findis.nf0	⊢ Ⅎ𝑥𝜓
bj-findis.nf1	⊢ Ⅎ𝑥𝜒
bj-findis.nfsuc	⊢ Ⅎ𝑥𝜃
bj-findis.0	⊢ (𝑥 = ∅ → (𝜓 → 𝜑))
bj-findis.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒))
bj-findis.suc	⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑))

Assertion

Ref	Expression
bj-findis	⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝜃(𝑥,𝑦)

Proof of Theorem bj-findis

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-nn0suc 10759	. . . . 5 ⊢ (𝑧 ∈ ω ↔ (𝑧 = ∅ ∨ ∃𝑦 ∈ ω 𝑧 = suc 𝑦))
2		pm3.21 260	. . . . . . . 8 ⊢ (𝜓 → (𝑧 = ∅ → (𝑧 = ∅ ∧ 𝜓)))
3	2	ad2antrr 471	. . . . . . 7 ⊢ (((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) ∧ ∀𝑦 ∈ 𝑧 (𝑦 ∈ ω → 𝜒)) → (𝑧 = ∅ → (𝑧 = ∅ ∧ 𝜓)))
4		pm2.04 81	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑧 → (𝑦 ∈ ω → 𝜒)) → (𝑦 ∈ ω → (𝑦 ∈ 𝑧 → 𝜒)))
5	4	ralimi2 2423	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑧 (𝑦 ∈ ω → 𝜒) → ∀𝑦 ∈ ω (𝑦 ∈ 𝑧 → 𝜒))
6		imim2 54	. . . . . . . . . . . 12 ⊢ ((𝜒 → 𝜃) → ((𝑦 ∈ 𝑧 → 𝜒) → (𝑦 ∈ 𝑧 → 𝜃)))
7	6	ral2imi 2427	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ ω (𝜒 → 𝜃) → (∀𝑦 ∈ ω (𝑦 ∈ 𝑧 → 𝜒) → ∀𝑦 ∈ ω (𝑦 ∈ 𝑧 → 𝜃)))
8	7	imp 122	. . . . . . . . . 10 ⊢ ((∀𝑦 ∈ ω (𝜒 → 𝜃) ∧ ∀𝑦 ∈ ω (𝑦 ∈ 𝑧 → 𝜒)) → ∀𝑦 ∈ ω (𝑦 ∈ 𝑧 → 𝜃))
9	5, 8	sylan2 280	. . . . . . . . 9 ⊢ ((∀𝑦 ∈ ω (𝜒 → 𝜃) ∧ ∀𝑦 ∈ 𝑧 (𝑦 ∈ ω → 𝜒)) → ∀𝑦 ∈ ω (𝑦 ∈ 𝑧 → 𝜃))
10		r19.29 2494	. . . . . . . . . . 11 ⊢ ((∀𝑦 ∈ ω (𝑦 ∈ 𝑧 → 𝜃) ∧ ∃𝑦 ∈ ω 𝑧 = suc 𝑦) → ∃𝑦 ∈ ω ((𝑦 ∈ 𝑧 → 𝜃) ∧ 𝑧 = suc 𝑦))
11		vex 2604	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
12	11	sucid 4172	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ suc 𝑦
13		eleq2 2142	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = suc 𝑦 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ suc 𝑦))
14	12, 13	mpbiri 166	. . . . . . . . . . . . . 14 ⊢ (𝑧 = suc 𝑦 → 𝑦 ∈ 𝑧)
15		ax-1 5	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = suc 𝑦 → ((𝑦 ∈ 𝑧 → 𝜃) → 𝑧 = suc 𝑦))
16		pm2.27 39	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝑧 → ((𝑦 ∈ 𝑧 → 𝜃) → 𝜃))
17	15, 16	anim12ii 335	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = suc 𝑦 ∧ 𝑦 ∈ 𝑧) → ((𝑦 ∈ 𝑧 → 𝜃) → (𝑧 = suc 𝑦 ∧ 𝜃)))
18	14, 17	mpdan 412	. . . . . . . . . . . . 13 ⊢ (𝑧 = suc 𝑦 → ((𝑦 ∈ 𝑧 → 𝜃) → (𝑧 = suc 𝑦 ∧ 𝜃)))
19	18	impcom 123	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ 𝑧 → 𝜃) ∧ 𝑧 = suc 𝑦) → (𝑧 = suc 𝑦 ∧ 𝜃))
20	19	reximi 2458	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ ω ((𝑦 ∈ 𝑧 → 𝜃) ∧ 𝑧 = suc 𝑦) → ∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃))
21	10, 20	syl 14	. . . . . . . . . 10 ⊢ ((∀𝑦 ∈ ω (𝑦 ∈ 𝑧 → 𝜃) ∧ ∃𝑦 ∈ ω 𝑧 = suc 𝑦) → ∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃))
22	21	ex 113	. . . . . . . . 9 ⊢ (∀𝑦 ∈ ω (𝑦 ∈ 𝑧 → 𝜃) → (∃𝑦 ∈ ω 𝑧 = suc 𝑦 → ∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃)))
23	9, 22	syl 14	. . . . . . . 8 ⊢ ((∀𝑦 ∈ ω (𝜒 → 𝜃) ∧ ∀𝑦 ∈ 𝑧 (𝑦 ∈ ω → 𝜒)) → (∃𝑦 ∈ ω 𝑧 = suc 𝑦 → ∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃)))
24	23	adantll 459	. . . . . . 7 ⊢ (((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) ∧ ∀𝑦 ∈ 𝑧 (𝑦 ∈ ω → 𝜒)) → (∃𝑦 ∈ ω 𝑧 = suc 𝑦 → ∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃)))
25	3, 24	orim12d 732	. . . . . 6 ⊢ (((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) ∧ ∀𝑦 ∈ 𝑧 (𝑦 ∈ ω → 𝜒)) → ((𝑧 = ∅ ∨ ∃𝑦 ∈ ω 𝑧 = suc 𝑦) → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃))))
26	25	ex 113	. . . . 5 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → (∀𝑦 ∈ 𝑧 (𝑦 ∈ ω → 𝜒) → ((𝑧 = ∅ ∨ ∃𝑦 ∈ ω 𝑧 = suc 𝑦) → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃)))))
27	1, 26	syl7bi 163	. . . 4 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → (∀𝑦 ∈ 𝑧 (𝑦 ∈ ω → 𝜒) → (𝑧 ∈ ω → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃)))))
28	27	alrimiv 1795	. . 3 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑧(∀𝑦 ∈ 𝑧 (𝑦 ∈ ω → 𝜒) → (𝑧 ∈ ω → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃)))))
29		nfv 1461	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ ω
30		bj-findis.nf1	. . . . 5 ⊢ Ⅎ𝑥𝜒
31	29, 30	nfim 1504	. . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ ω → 𝜒)
32		nfv 1461	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ ω
33		nfv 1461	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = ∅
34		bj-findis.nf0	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
35	33, 34	nfan 1497	. . . . . 6 ⊢ Ⅎ𝑥(𝑧 = ∅ ∧ 𝜓)
36		nfcv 2219	. . . . . . 7 ⊢ Ⅎ𝑥ω
37		nfv 1461	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 = suc 𝑦
38		bj-findis.nfsuc	. . . . . . . 8 ⊢ Ⅎ𝑥𝜃
39	37, 38	nfan 1497	. . . . . . 7 ⊢ Ⅎ𝑥(𝑧 = suc 𝑦 ∧ 𝜃)
40	36, 39	nfrexxy 2403	. . . . . 6 ⊢ Ⅎ𝑥∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃)
41	35, 40	nfor 1506	. . . . 5 ⊢ Ⅎ𝑥((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃))
42	32, 41	nfim 1504	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ ω → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃)))
43		nfv 1461	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ ω → 𝜑)
44		nfv 1461	. . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ ω → 𝜒)
45		eleq1 2141	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω))
46	45	biimprd 156	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ ω → 𝑥 ∈ ω))
47		bj-findis.1	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒))
48	46, 47	imim12d 73	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ω → 𝜑) → (𝑦 ∈ ω → 𝜒)))
49		eleq1 2141	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ ω ↔ 𝑧 ∈ ω))
50	49	biimpd 142	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ ω → 𝑧 ∈ ω))
51		eqtr 2098	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑧 = ∅) → 𝑥 = ∅)
52		bj-findis.0	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝜓 → 𝜑))
53	51, 52	syl 14	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑧 = ∅) → (𝜓 → 𝜑))
54	53	expimpd 355	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑧 = ∅ ∧ 𝜓) → 𝜑))
55		eqtr 2098	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑧 = suc 𝑦) → 𝑥 = suc 𝑦)
56		bj-findis.suc	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑))
57	55, 56	syl 14	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑧 = suc 𝑦) → (𝜃 → 𝜑))
58	57	expimpd 355	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑧 = suc 𝑦 ∧ 𝜃) → 𝜑))
59	58	rexlimdvw 2480	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃) → 𝜑))
60	54, 59	jaod 669	. . . . 5 ⊢ (𝑥 = 𝑧 → (((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃)) → 𝜑))
61	50, 60	imim12d 73	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑧 ∈ ω → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃))) → (𝑥 ∈ ω → 𝜑)))
62	31, 42, 43, 44, 48, 61	setindis 10762	. . 3 ⊢ (∀𝑧(∀𝑦 ∈ 𝑧 (𝑦 ∈ ω → 𝜒) → (𝑧 ∈ ω → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃)))) → ∀𝑥(𝑥 ∈ ω → 𝜑))
63	28, 62	syl 14	. 2 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥(𝑥 ∈ ω → 𝜑))
64		df-ral 2353	. 2 ⊢ (∀𝑥 ∈ ω 𝜑 ↔ ∀𝑥(𝑥 ∈ ω → 𝜑))
65	63, 64	sylibr 132	1 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑)

Syntax hints:   → wi 4   ∧ wa 102   ∨ wo 661  ∀wal 1282   = wceq 1284  Ⅎwnf 1389   ∈ wcel 1433  ∀wral 2348  ∃wrex 2349  ∅c0 3251  suc csuc 4120  ωcom 4331

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-nul 3904  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-bd0 10604  ax-bdim 10605  ax-bdan 10606  ax-bdor 10607  ax-bdn 10608  ax-bdal 10609  ax-bdex 10610  ax-bdeq 10611  ax-bdel 10612  ax-bdsb 10613  ax-bdsep 10675  ax-infvn 10736

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-sn 3404  df-pr 3405  df-uni 3602  df-int 3637  df-suc 4126  df-iom 4332  df-bdc 10632  df-bj-ind 10722

This theorem is referenced by:  bj-findisg  10775  bj-findes  10776