Theorem bj-bdfindis 13829

Description: Bounded induction (principle of induction for bounded formulas), using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See finds 4577 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4577, finds2 4578, finds1 4579. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
bj-bdfindis	⊢ ((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-bdfindis

Step	Hyp	Ref	Expression
1		bj-bdfindis.nf0	. . . 4 ⊢ Ⅎ𝑥𝜓
2		0ex 4109	. . . 4 ⊢ ∅ ∈ V
3		bj-bdfindis.0	. . . 4 ⊢ (𝑥 = ∅ → (𝜓 → 𝜑))
4	1, 2, 3	elabf2 13663	. . 3 ⊢ (𝜓 → ∅ ∈ {𝑥 ∣ 𝜑})
5		bj-bdfindis.nf1	. . . . . 6 ⊢ Ⅎ𝑥𝜒
6		bj-bdfindis.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒))
7	5, 6	elabf1 13662	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → 𝜒)
8		bj-bdfindis.nfsuc	. . . . . 6 ⊢ Ⅎ𝑥𝜃
9		vex 2729	. . . . . . 7 ⊢ 𝑦 ∈ V
10	9	bj-sucex 13805	. . . . . 6 ⊢ suc 𝑦 ∈ V
11		bj-bdfindis.suc	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑))
12	8, 10, 11	elabf2 13663	. . . . 5 ⊢ (𝜃 → suc 𝑦 ∈ {𝑥 ∣ 𝜑})
13	7, 12	imim12i 59	. . . 4 ⊢ ((𝜒 → 𝜃) → (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑}))
14	13	ralimi 2529	. . 3 ⊢ (∀𝑦 ∈ ω (𝜒 → 𝜃) → ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑}))
15		bj-bdfindis.bd	. . . . 5 ⊢ BOUNDED 𝜑
16	15	bdcab 13731	. . . 4 ⊢ BOUNDED {𝑥 ∣ 𝜑}
17	16	bdpeano5 13825	. . 3 ⊢ ((∅ ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})) → ω ⊆ {𝑥 ∣ 𝜑})
18	4, 14, 17	syl2an 287	. 2 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ω ⊆ {𝑥 ∣ 𝜑})
19		ssabral 3213	. 2 ⊢ (ω ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥 ∈ ω 𝜑)
20	18, 19	sylib 121	1 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343  Ⅎwnf 1448   ∈ wcel 2136  {cab 2151  ∀wral 2444   ⊆ wss 3116  ∅c0 3409  suc csuc 4343  ωcom 4567  BOUNDED wbd 13694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-nul 4108  ax-pr 4187  ax-un 4411  ax-bd0 13695  ax-bdor 13698  ax-bdex 13701  ax-bdeq 13702  ax-bdel 13703  ax-bdsb 13704  ax-bdsep 13766  ax-infvn 13823

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-sn 3582  df-pr 3583  df-uni 3790  df-int 3825  df-suc 4349  df-iom 4568  df-bdc 13723  df-bj-ind 13809

This theorem is referenced by:  bj-bdfindisg  13830  bj-bdfindes  13831  bj-nn0suc0  13832