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Theorem bj-bdfindis 13134
 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 4509 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4509, finds2 4510, finds1 4511. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-bdfindis.bd BOUNDED 𝜑
bj-bdfindis.nf0 𝑥𝜓
bj-bdfindis.nf1 𝑥𝜒
bj-bdfindis.nfsuc 𝑥𝜃
bj-bdfindis.0 (𝑥 = ∅ → (𝜓𝜑))
bj-bdfindis.1 (𝑥 = 𝑦 → (𝜑𝜒))
bj-bdfindis.suc (𝑥 = suc 𝑦 → (𝜃𝜑))
Assertion
Ref Expression
bj-bdfindis ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ∀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
StepHypRef Expression
1 bj-bdfindis.nf0 . . . 4 𝑥𝜓
2 0ex 4050 . . . 4 ∅ ∈ V
3 bj-bdfindis.0 . . . 4 (𝑥 = ∅ → (𝜓𝜑))
41, 2, 3elabf2 12978 . . 3 (𝜓 → ∅ ∈ {𝑥𝜑})
5 bj-bdfindis.nf1 . . . . . 6 𝑥𝜒
6 bj-bdfindis.1 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜒))
75, 6elabf1 12977 . . . . 5 (𝑦 ∈ {𝑥𝜑} → 𝜒)
8 bj-bdfindis.nfsuc . . . . . 6 𝑥𝜃
9 vex 2684 . . . . . . 7 𝑦 ∈ V
109bj-sucex 13110 . . . . . 6 suc 𝑦 ∈ V
11 bj-bdfindis.suc . . . . . 6 (𝑥 = suc 𝑦 → (𝜃𝜑))
128, 10, 11elabf2 12978 . . . . 5 (𝜃 → suc 𝑦 ∈ {𝑥𝜑})
137, 12imim12i 59 . . . 4 ((𝜒𝜃) → (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑}))
1413ralimi 2493 . . 3 (∀𝑦 ∈ ω (𝜒𝜃) → ∀𝑦 ∈ ω (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑}))
15 bj-bdfindis.bd . . . . 5 BOUNDED 𝜑
1615bdcab 13036 . . . 4 BOUNDED {𝑥𝜑}
1716bdpeano5 13130 . . 3 ((∅ ∈ {𝑥𝜑} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑})) → ω ⊆ {𝑥𝜑})
184, 14, 17syl2an 287 . 2 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ω ⊆ {𝑥𝜑})
19 ssabral 3163 . 2 (ω ⊆ {𝑥𝜑} ↔ ∀𝑥 ∈ ω 𝜑)
2018, 19sylib 121 1 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ∀𝑥 ∈ ω 𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331  Ⅎwnf 1436   ∈ wcel 1480  {cab 2123  ∀wral 2414   ⊆ wss 3066  ∅c0 3358  suc csuc 4282  ωcom 4499  BOUNDED wbd 12999 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-nul 4049  ax-pr 4126  ax-un 4350  ax-bd0 13000  ax-bdor 13003  ax-bdex 13006  ax-bdeq 13007  ax-bdel 13008  ax-bdsb 13009  ax-bdsep 13071  ax-infvn 13128 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-sn 3528  df-pr 3529  df-uni 3732  df-int 3767  df-suc 4288  df-iom 4500  df-bdc 13028  df-bj-ind 13114 This theorem is referenced by:  bj-bdfindisg  13135  bj-bdfindes  13136  bj-nn0suc0  13137
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