Theorem bj-bdfindis 15883

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 4648 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4648, finds2 4649, finds1 4650. (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 4171	. . . 4 ⊢ ∅ ∈ V
3		bj-bdfindis.0	. . . 4 ⊢ (𝑥 = ∅ → (𝜓 → 𝜑))
4	1, 2, 3	elabf2 15718	. . 3 ⊢ (𝜓 → ∅ ∈ {𝑥 ∣ 𝜑})
5		bj-bdfindis.nf1	. . . . . 6 ⊢ Ⅎ𝑥𝜒
6		bj-bdfindis.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒))
7	5, 6	elabf1 15717	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → 𝜒)
8		bj-bdfindis.nfsuc	. . . . . 6 ⊢ Ⅎ𝑥𝜃
9		vex 2775	. . . . . . 7 ⊢ 𝑦 ∈ V
10	9	bj-sucex 15859	. . . . . 6 ⊢ suc 𝑦 ∈ V
11		bj-bdfindis.suc	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑))
12	8, 10, 11	elabf2 15718	. . . . 5 ⊢ (𝜃 → suc 𝑦 ∈ {𝑥 ∣ 𝜑})
13	7, 12	imim12i 59	. . . 4 ⊢ ((𝜒 → 𝜃) → (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑}))
14	13	ralimi 2569	. . 3 ⊢ (∀𝑦 ∈ ω (𝜒 → 𝜃) → ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑}))
15		bj-bdfindis.bd	. . . . 5 ⊢ BOUNDED 𝜑
16	15	bdcab 15785	. . . 4 ⊢ BOUNDED {𝑥 ∣ 𝜑}
17	16	bdpeano5 15879	. . 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 3264	. 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 1373  Ⅎwnf 1483   ∈ wcel 2176  {cab 2191  ∀wral 2484   ⊆ wss 3166  ∅c0 3460  suc csuc 4412  ωcom 4638  BOUNDED wbd 15748

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-nul 4170  ax-pr 4253  ax-un 4480  ax-bd0 15749  ax-bdor 15752  ax-bdex 15755  ax-bdeq 15756  ax-bdel 15757  ax-bdsb 15758  ax-bdsep 15820  ax-infvn 15877

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-suc 4418  df-iom 4639  df-bdc 15777  df-bj-ind 15863

This theorem is referenced by:  bj-bdfindisg  15884  bj-bdfindes  15885  bj-nn0suc0  15886