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Theorem bj-bdfindis 14581
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 4599 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4599, finds2 4600, finds1 4601. (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 4130 . . . 4 ∅ ∈ V
3 bj-bdfindis.0 . . . 4 (𝑥 = ∅ → (𝜓𝜑))
41, 2, 3elabf2 14416 . . 3 (𝜓 → ∅ ∈ {𝑥𝜑})
5 bj-bdfindis.nf1 . . . . . 6 𝑥𝜒
6 bj-bdfindis.1 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜒))
75, 6elabf1 14415 . . . . 5 (𝑦 ∈ {𝑥𝜑} → 𝜒)
8 bj-bdfindis.nfsuc . . . . . 6 𝑥𝜃
9 vex 2740 . . . . . . 7 𝑦 ∈ V
109bj-sucex 14557 . . . . . 6 suc 𝑦 ∈ V
11 bj-bdfindis.suc . . . . . 6 (𝑥 = suc 𝑦 → (𝜃𝜑))
128, 10, 11elabf2 14416 . . . . 5 (𝜃 → suc 𝑦 ∈ {𝑥𝜑})
137, 12imim12i 59 . . . 4 ((𝜒𝜃) → (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑}))
1413ralimi 2540 . . 3 (∀𝑦 ∈ ω (𝜒𝜃) → ∀𝑦 ∈ ω (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑}))
15 bj-bdfindis.bd . . . . 5 BOUNDED 𝜑
1615bdcab 14483 . . . 4 BOUNDED {𝑥𝜑}
1716bdpeano5 14577 . . 3 ((∅ ∈ {𝑥𝜑} ∧ ∀𝑦 ∈ ω (𝑦 ∈ {𝑥𝜑} → suc 𝑦 ∈ {𝑥𝜑})) → ω ⊆ {𝑥𝜑})
184, 14, 17syl2an 289 . 2 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ω ⊆ {𝑥𝜑})
19 ssabral 3226 . 2 (ω ⊆ {𝑥𝜑} ↔ ∀𝑥 ∈ ω 𝜑)
2018, 19sylib 122 1 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ∀𝑥 ∈ ω 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wnf 1460  wcel 2148  {cab 2163  wral 2455  wss 3129  c0 3422  suc csuc 4365  ωcom 4589  BOUNDED wbd 14446
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4129  ax-pr 4209  ax-un 4433  ax-bd0 14447  ax-bdor 14450  ax-bdex 14453  ax-bdeq 14454  ax-bdel 14455  ax-bdsb 14456  ax-bdsep 14518  ax-infvn 14575
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-suc 4371  df-iom 4590  df-bdc 14475  df-bj-ind 14561
This theorem is referenced by:  bj-bdfindisg  14582  bj-bdfindes  14583  bj-nn0suc0  14584
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