Theorem bj-bdfindis 13982

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 4584 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4584, finds2 4585, finds1 4586. (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 4116	. . . 4 ⊢ ∅ ∈ V
3		bj-bdfindis.0	. . . 4 ⊢ (𝑥 = ∅ → (𝜓 → 𝜑))
4	1, 2, 3	elabf2 13817	. . 3 ⊢ (𝜓 → ∅ ∈ {𝑥 ∣ 𝜑})
5		bj-bdfindis.nf1	. . . . . 6 ⊢ Ⅎ𝑥𝜒
6		bj-bdfindis.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒))
7	5, 6	elabf1 13816	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → 𝜒)
8		bj-bdfindis.nfsuc	. . . . . 6 ⊢ Ⅎ𝑥𝜃
9		vex 2733	. . . . . . 7 ⊢ 𝑦 ∈ V
10	9	bj-sucex 13958	. . . . . 6 ⊢ suc 𝑦 ∈ V
11		bj-bdfindis.suc	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑))
12	8, 10, 11	elabf2 13817	. . . . 5 ⊢ (𝜃 → suc 𝑦 ∈ {𝑥 ∣ 𝜑})
13	7, 12	imim12i 59	. . . 4 ⊢ ((𝜒 → 𝜃) → (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑}))
14	13	ralimi 2533	. . 3 ⊢ (∀𝑦 ∈ ω (𝜒 → 𝜃) → ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑}))
15		bj-bdfindis.bd	. . . . 5 ⊢ BOUNDED 𝜑
16	15	bdcab 13884	. . . 4 ⊢ BOUNDED {𝑥 ∣ 𝜑}
17	16	bdpeano5 13978	. . 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 3218	. 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 1348  Ⅎwnf 1453   ∈ wcel 2141  {cab 2156  ∀wral 2448   ⊆ wss 3121  ∅c0 3414  suc csuc 4350  ωcom 4574  BOUNDED wbd 13847

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-nul 4115  ax-pr 4194  ax-un 4418  ax-bd0 13848  ax-bdor 13851  ax-bdex 13854  ax-bdeq 13855  ax-bdel 13856  ax-bdsb 13857  ax-bdsep 13919  ax-infvn 13976

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-suc 4356  df-iom 4575  df-bdc 13876  df-bj-ind 13962

This theorem is referenced by:  bj-bdfindisg  13983  bj-bdfindes  13984  bj-nn0suc0  13985