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Theorem bj-bdfindis 7169
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 4250 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4250, finds2 4251, finds1 4252. (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 3858 . . . 4 V
3 bj-bdfindis.0 . . . 4 (x = ∅ → (ψφ))
41, 2, 3elabf2 7028 . . 3 (ψ → ∅ {xφ})
5 bj-bdfindis.nf1 . . . . . 6 xχ
6 bj-bdfindis.1 . . . . . 6 (x = y → (φχ))
75, 6elabf1 7027 . . . . 5 (y {xφ} → χ)
8 bj-bdfindis.nfsuc . . . . . 6 xθ
9 vex 2538 . . . . . . 7 y V
109bj-sucex 7146 . . . . . 6 suc y V
11 bj-bdfindis.suc . . . . . 6 (x = suc y → (θφ))
128, 10, 11elabf2 7028 . . . . 5 (θ → suc y {xφ})
137, 12imim12i 53 . . . 4 ((χθ) → (y {xφ} → suc y {xφ}))
1413ralimi 2362 . . 3 (y 𝜔 (χθ) → y 𝜔 (y {xφ} → suc y {xφ}))
15 bj-bdfindis.bd . . . . 5 BOUNDED φ
1615bdcab 7076 . . . 4 BOUNDED {xφ}
1716bdpeano5 7165 . . 3 ((∅ {xφ} y 𝜔 (y {xφ} → suc y {xφ})) → 𝜔 ⊆ {xφ})
184, 14, 17syl2an 273 . 2 ((ψ y 𝜔 (χθ)) → 𝜔 ⊆ {xφ})
19 ssabral 2988 . 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 1228  wnf 1329   wcel 1374  {cab 2008  wral 2284  wss 2894  c0 3201  suc csuc 4051  𝜔com 4240  BOUNDED wbd 7039
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3857  ax-pr 3918  ax-un 4120  ax-bd0 7040  ax-bdor 7043  ax-bdex 7046  ax-bdeq 7047  ax-bdel 7048  ax-bdsb 7049  ax-bdsep 7111  ax-infvn 7163
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241  df-bdc 7068  df-bj-ind 7150
This theorem is referenced by:  bj-bdfindisg  7170  bj-bdfindes  7171  bj-nn0suc0  7172
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