Theorem bj-bdfindis 15593

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 4636 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4636, finds2 4637, finds1 4638. (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 4160	. . . 4 ⊢ ∅ ∈ V
3		bj-bdfindis.0	. . . 4 ⊢ (𝑥 = ∅ → (𝜓 → 𝜑))
4	1, 2, 3	elabf2 15428	. . 3 ⊢ (𝜓 → ∅ ∈ {𝑥 ∣ 𝜑})
5		bj-bdfindis.nf1	. . . . . 6 ⊢ Ⅎ𝑥𝜒
6		bj-bdfindis.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒))
7	5, 6	elabf1 15427	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → 𝜒)
8		bj-bdfindis.nfsuc	. . . . . 6 ⊢ Ⅎ𝑥𝜃
9		vex 2766	. . . . . . 7 ⊢ 𝑦 ∈ V
10	9	bj-sucex 15569	. . . . . 6 ⊢ suc 𝑦 ∈ V
11		bj-bdfindis.suc	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑))
12	8, 10, 11	elabf2 15428	. . . . 5 ⊢ (𝜃 → suc 𝑦 ∈ {𝑥 ∣ 𝜑})
13	7, 12	imim12i 59	. . . 4 ⊢ ((𝜒 → 𝜃) → (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑}))
14	13	ralimi 2560	. . 3 ⊢ (∀𝑦 ∈ ω (𝜒 → 𝜃) → ∀𝑦 ∈ ω (𝑦 ∈ {𝑥 ∣ 𝜑} → suc 𝑦 ∈ {𝑥 ∣ 𝜑}))
15		bj-bdfindis.bd	. . . . 5 ⊢ BOUNDED 𝜑
16	15	bdcab 15495	. . . 4 ⊢ BOUNDED {𝑥 ∣ 𝜑}
17	16	bdpeano5 15589	. . 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 3254	. 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 1364  Ⅎwnf 1474   ∈ wcel 2167  {cab 2182  ∀wral 2475   ⊆ wss 3157  ∅c0 3450  suc csuc 4400  ωcom 4626  BOUNDED wbd 15458

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 4159  ax-pr 4242  ax-un 4468  ax-bd0 15459  ax-bdor 15462  ax-bdex 15465  ax-bdeq 15466  ax-bdel 15467  ax-bdsb 15468  ax-bdsep 15530  ax-infvn 15587

This theorem depends on definitions:  df-bi 117  df-tru 1367  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 3451  df-sn 3628  df-pr 3629  df-uni 3840  df-int 3875  df-suc 4406  df-iom 4627  df-bdc 15487  df-bj-ind 15573

This theorem is referenced by:  bj-bdfindisg  15594  bj-bdfindes  15595  bj-nn0suc0  15596