Theorem bj-bdfindis 15845

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 4647 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4647, finds2 4648, finds1 4649. (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 4170	. . . 4 ⊢ ∅ ∈ V
3		bj-bdfindis.0	. . . 4 ⊢ (𝑥 = ∅ → (𝜓 → 𝜑))
4	1, 2, 3	elabf2 15680	. . 3 ⊢ (𝜓 → ∅ ∈ {𝑥 ∣ 𝜑})
5		bj-bdfindis.nf1	. . . . . 6 ⊢ Ⅎ𝑥𝜒
6		bj-bdfindis.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒))
7	5, 6	elabf1 15679	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → 𝜒)
8		bj-bdfindis.nfsuc	. . . . . 6 ⊢ Ⅎ𝑥𝜃
9		vex 2774	. . . . . . 7 ⊢ 𝑦 ∈ V
10	9	bj-sucex 15821	. . . . . 6 ⊢ suc 𝑦 ∈ V
11		bj-bdfindis.suc	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑))
12	8, 10, 11	elabf2 15680	. . . . 5 ⊢ (𝜃 → suc 𝑦 ∈ {𝑥 ∣ 𝜑})
13	7, 12	imim12i 59	. . . 4 ⊢ ((𝜒 → 𝜃) → (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑}))
14	13	ralimi 2568	. . 3 ⊢ (∀𝑦 ∈ ω (𝜒 → 𝜃) → ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑}))
15		bj-bdfindis.bd	. . . . 5 ⊢ BOUNDED 𝜑
16	15	bdcab 15747	. . . 4 ⊢ BOUNDED {𝑥 ∣ 𝜑}
17	16	bdpeano5 15841	. . 3 ⊢ ((∅ ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑})) → ω ⊆ {𝑥 ∣ 𝜑})
18	4, 14, 17	syl2an 289	. 2 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ω ⊆ {𝑥 ∣ 𝜑})
19		ssabral 3263	. 2 ⊢ (ω ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥 ∈ ω 𝜑)
20	18, 19	sylib 122	1 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1372  Ⅎwnf 1482   ∈ wcel 2175  {cab 2190  ∀wral 2483   ⊆ wss 3165  ∅c0 3459  suc csuc 4411  ωcom 4637  BOUNDED wbd 15710

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-nul 4169  ax-pr 4252  ax-un 4479  ax-bd0 15711  ax-bdor 15714  ax-bdex 15717  ax-bdeq 15718  ax-bdel 15719  ax-bdsb 15720  ax-bdsep 15782  ax-infvn 15839

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-sn 3638  df-pr 3639  df-uni 3850  df-int 3885  df-suc 4417  df-iom 4638  df-bdc 15739  df-bj-ind 15825

This theorem is referenced by:  bj-bdfindisg  15846  bj-bdfindes  15847  bj-nn0suc0  15848