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Theorem bj-bdfindis 9407
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 4266 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4266, finds2 4267, finds1 4268. (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 3875 . . . 4 V
3 bj-bdfindis.0 . . . 4 (x = ∅ → (ψφ))
41, 2, 3elabf2 9256 . . 3 (ψ → ∅ {xφ})
5 bj-bdfindis.nf1 . . . . . 6 xχ
6 bj-bdfindis.1 . . . . . 6 (x = y → (φχ))
75, 6elabf1 9255 . . . . 5 (y {xφ} → χ)
8 bj-bdfindis.nfsuc . . . . . 6 xθ
9 vex 2554 . . . . . . 7 y V
109bj-sucex 9378 . . . . . 6 suc y V
11 bj-bdfindis.suc . . . . . 6 (x = suc y → (θφ))
128, 10, 11elabf2 9256 . . . . 5 (θ → suc y {xφ})
137, 12imim12i 53 . . . 4 ((χθ) → (y {xφ} → suc y {xφ}))
1413ralimi 2378 . . 3 (y 𝜔 (χθ) → y 𝜔 (y {xφ} → suc y {xφ}))
15 bj-bdfindis.bd . . . . 5 BOUNDED φ
1615bdcab 9304 . . . 4 BOUNDED {xφ}
1716bdpeano5 9403 . . 3 ((∅ {xφ} y 𝜔 (y {xφ} → suc y {xφ})) → 𝜔 ⊆ {xφ})
184, 14, 17syl2an 273 . 2 ((ψ y 𝜔 (χθ)) → 𝜔 ⊆ {xφ})
19 ssabral 3005 . 2 (𝜔 ⊆ {xφ} ↔ x 𝜔 φ)
2018, 19sylib 127 1 ((ψ y 𝜔 (χθ)) → x 𝜔 φ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wnf 1346   wcel 1390  {cab 2023  wral 2300  wss 2911  c0 3218  suc csuc 4068  𝜔com 4256  BOUNDED wbd 9267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874  ax-pr 3935  ax-un 4136  ax-bd0 9268  ax-bdor 9271  ax-bdex 9274  ax-bdeq 9275  ax-bdel 9276  ax-bdsb 9277  ax-bdsep 9339  ax-infvn 9401
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257  df-bdc 9296  df-bj-ind 9386
This theorem is referenced by:  bj-bdfindisg  9408  bj-bdfindes  9409  bj-nn0suc0  9410
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