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Theorem bj-bdfindis 13522
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 4559 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4559, finds2 4560, finds1 4561. (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 4091 . . . 4 ∅ ∈ V
3 bj-bdfindis.0 . . . 4 (𝑥 = ∅ → (𝜓𝜑))
41, 2, 3elabf2 13356 . . 3 (𝜓 → ∅ ∈ {𝑥𝜑})
5 bj-bdfindis.nf1 . . . . . 6 𝑥𝜒
6 bj-bdfindis.1 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜒))
75, 6elabf1 13355 . . . . 5 (𝑦 ∈ {𝑥𝜑} → 𝜒)
8 bj-bdfindis.nfsuc . . . . . 6 𝑥𝜃
9 vex 2715 . . . . . . 7 𝑦 ∈ V
109bj-sucex 13498 . . . . . 6 suc 𝑦 ∈ V
11 bj-bdfindis.suc . . . . . 6 (𝑥 = suc 𝑦 → (𝜃𝜑))
128, 10, 11elabf2 13356 . . . . 5 (𝜃 → suc 𝑦 ∈ {𝑥𝜑})
137, 12imim12i 59 . . . 4 ((𝜒𝜃) → (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑}))
1413ralimi 2520 . . 3 (∀𝑦 ∈ ω (𝜒𝜃) → ∀𝑦 ∈ ω (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑}))
15 bj-bdfindis.bd . . . . 5 BOUNDED 𝜑
1615bdcab 13424 . . . 4 BOUNDED {𝑥𝜑}
1716bdpeano5 13518 . . 3 ((∅ ∈ {𝑥𝜑} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑})) → ω ⊆ {𝑥𝜑})
184, 14, 17syl2an 287 . 2 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ω ⊆ {𝑥𝜑})
19 ssabral 3199 . 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 1335  wnf 1440  wcel 2128  {cab 2143  wral 2435  wss 3102  c0 3394  suc csuc 4325  ωcom 4549  BOUNDED wbd 13387
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-nul 4090  ax-pr 4169  ax-un 4393  ax-bd0 13388  ax-bdor 13391  ax-bdex 13394  ax-bdeq 13395  ax-bdel 13396  ax-bdsb 13397  ax-bdsep 13459  ax-infvn 13516
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-sn 3566  df-pr 3567  df-uni 3773  df-int 3808  df-suc 4331  df-iom 4550  df-bdc 13416  df-bj-ind 13502
This theorem is referenced by:  bj-bdfindisg  13523  bj-bdfindes  13524  bj-nn0suc0  13525
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