Theorem bj-findis 15709

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 15677 for a bounded version not requiring ax-setind 4574. See finds 4637 for a proof in IZF. From this version, it is easy to prove of finds 4637, finds2 4638, finds1 4639. (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 15694	. . . . 5 ⊢ (𝑧 ∈ ω ↔ (𝑧 = ∅ ∨ ∃𝑦 ∈ ω 𝑧 = suc 𝑦))
2		pm3.21 264	. . . . . . . 8 ⊢ (𝜓 → (𝑧 = ∅ → (𝑧 = ∅ ∧ 𝜓)))
3	2	ad2antrr 488	. . . . . . 7 ⊢ (((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) ∧ ∀𝑦 ∈ 𝑧 (𝑦 ∈ ω → 𝜒)) → (𝑧 = ∅ → (𝑧 = ∅ ∧ 𝜓)))
4		pm2.04 82	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑧 → (𝑦 ∈ ω → 𝜒)) → (𝑦 ∈ ω → (𝑦 ∈ 𝑧 → 𝜒)))
5	4	ralimi2 2557	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑧 (𝑦 ∈ ω → 𝜒) → ∀𝑦 ∈ ω (𝑦 ∈ 𝑧 → 𝜒))
6		imim2 55	. . . . . . . . . . . 12 ⊢ ((𝜒 → 𝜃) → ((𝑦 ∈ 𝑧 → 𝜒) → (𝑦 ∈ 𝑧 → 𝜃)))
7	6	ral2imi 2562	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ ω (𝜒 → 𝜃) → (∀𝑦 ∈ ω (𝑦 ∈ 𝑧 → 𝜒) → ∀𝑦 ∈ ω (𝑦 ∈ 𝑧 → 𝜃)))
8	7	imp 124	. . . . . . . . . 10 ⊢ ((∀𝑦 ∈ ω (𝜒 → 𝜃) ∧ ∀𝑦 ∈ ω (𝑦 ∈ 𝑧 → 𝜒)) → ∀𝑦 ∈ ω (𝑦 ∈ 𝑧 → 𝜃))
9	5, 8	sylan2 286	. . . . . . . . 9 ⊢ ((∀𝑦 ∈ ω (𝜒 → 𝜃) ∧ ∀𝑦 ∈ 𝑧 (𝑦 ∈ ω → 𝜒)) → ∀𝑦 ∈ ω (𝑦 ∈ 𝑧 → 𝜃))
10		r19.29 2634	. . . . . . . . . . 11 ⊢ ((∀𝑦 ∈ ω (𝑦 ∈ 𝑧 → 𝜃) ∧ ∃𝑦 ∈ ω 𝑧 = suc 𝑦) → ∃𝑦 ∈ ω ((𝑦 ∈ 𝑧 → 𝜃) ∧ 𝑧 = suc 𝑦))
11		vex 2766	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
12	11	sucid 4453	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ suc 𝑦
13		eleq2 2260	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = suc 𝑦 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ suc 𝑦))
14	12, 13	mpbiri 168	. . . . . . . . . . . . . 14 ⊢ (𝑧 = suc 𝑦 → 𝑦 ∈ 𝑧)
15		ax-1 6	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = suc 𝑦 → ((𝑦 ∈ 𝑧 → 𝜃) → 𝑧 = suc 𝑦))
16		pm2.27 40	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝑧 → ((𝑦 ∈ 𝑧 → 𝜃) → 𝜃))
17	15, 16	anim12ii 343	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = suc 𝑦 ∧ 𝑦 ∈ 𝑧) → ((𝑦 ∈ 𝑧 → 𝜃) → (𝑧 = suc 𝑦 ∧ 𝜃)))
18	14, 17	mpdan 421	. . . . . . . . . . . . 13 ⊢ (𝑧 = suc 𝑦 → ((𝑦 ∈ 𝑧 → 𝜃) → (𝑧 = suc 𝑦 ∧ 𝜃)))
19	18	impcom 125	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ 𝑧 → 𝜃) ∧ 𝑧 = suc 𝑦) → (𝑧 = suc 𝑦 ∧ 𝜃))
20	19	reximi 2594	. . . . . . . . . . 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 115	. . . . . . . . 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 476	. . . . . . 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 787	. . . . . 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 115	. . . . 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 165	. . . 4 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → (∀𝑦 ∈ 𝑧 (𝑦 ∈ ω → 𝜒) → (𝑧 ∈ ω → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃)))))
28	27	alrimiv 1888	. . 3 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑧(∀𝑦 ∈ 𝑧 (𝑦 ∈ ω → 𝜒) → (𝑧 ∈ ω → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃)))))
29		nfv 1542	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ ω
30		bj-findis.nf1	. . . . 5 ⊢ Ⅎ𝑥𝜒
31	29, 30	nfim 1586	. . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ ω → 𝜒)
32		nfv 1542	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ ω
33		nfv 1542	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = ∅
34		bj-findis.nf0	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
35	33, 34	nfan 1579	. . . . . 6 ⊢ Ⅎ𝑥(𝑧 = ∅ ∧ 𝜓)
36		nfcv 2339	. . . . . . 7 ⊢ Ⅎ𝑥ω
37		nfv 1542	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 = suc 𝑦
38		bj-findis.nfsuc	. . . . . . . 8 ⊢ Ⅎ𝑥𝜃
39	37, 38	nfan 1579	. . . . . . 7 ⊢ Ⅎ𝑥(𝑧 = suc 𝑦 ∧ 𝜃)
40	36, 39	nfrexw 2536	. . . . . 6 ⊢ Ⅎ𝑥∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃)
41	35, 40	nfor 1588	. . . . 5 ⊢ Ⅎ𝑥((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃))
42	32, 41	nfim 1586	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ ω → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃)))
43		nfv 1542	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ ω → 𝜑)
44		nfv 1542	. . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ ω → 𝜒)
45		eleq1 2259	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω))
46	45	biimprd 158	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ ω → 𝑥 ∈ ω))
47		bj-findis.1	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒))
48	46, 47	imim12d 74	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ω → 𝜑) → (𝑦 ∈ ω → 𝜒)))
49		eleq1 2259	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ ω ↔ 𝑧 ∈ ω))
50	49	biimpd 144	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ ω → 𝑧 ∈ ω))
51		eqtr 2214	. . . . . . . 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 363	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑧 = ∅ ∧ 𝜓) → 𝜑))
55		eqtr 2214	. . . . . . . . 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 363	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑧 = suc 𝑦 ∧ 𝜃) → 𝜑))
59	58	rexlimdvw 2618	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃) → 𝜑))
60	54, 59	jaod 718	. . . . 5 ⊢ (𝑥 = 𝑧 → (((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃)) → 𝜑))
61	50, 60	imim12d 74	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑧 ∈ ω → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦 ∧ 𝜃))) → (𝑥 ∈ ω → 𝜑)))
62	31, 42, 43, 44, 48, 61	setindis 15697	. . 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 2480	. 2 ⊢ (∀𝑥 ∈ ω 𝜑 ↔ ∀𝑥(𝑥 ∈ ω → 𝜑))
65	63, 64	sylibr 134	1 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709  ∀wal 1362   = wceq 1364  Ⅎwnf 1474   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  ∅c0 3451  suc csuc 4401  ωcom 4627

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-nul 4160  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-bd0 15543  ax-bdim 15544  ax-bdan 15545  ax-bdor 15546  ax-bdn 15547  ax-bdal 15548  ax-bdex 15549  ax-bdeq 15550  ax-bdel 15551  ax-bdsb 15552  ax-bdsep 15614  ax-infvn 15671

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-sn 3629  df-pr 3630  df-uni 3841  df-int 3876  df-suc 4407  df-iom 4628  df-bdc 15571  df-bj-ind 15657

This theorem is referenced by:  bj-findisg  15710  bj-findes  15711