Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-bdfindes Structured version   GIF version

Theorem bj-bdfindes 8405
Description: Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 8403 for explanations. From this version, it is easy to prove the bounded version of findes 4242. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-bdfindes.bd BOUNDED φ
Assertion
Ref Expression
bj-bdfindes (([∅ / x]φ x 𝜔 (φ[suc x / x]φ)) → x 𝜔 φ)

Proof of Theorem bj-bdfindes
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 nfv 1395 . . . 4 yφ
2 nfv 1395 . . . 4 y[suc x / x]φ
31, 2nfim 1438 . . 3 y(φ[suc x / x]φ)
4 nfs1v 1789 . . . 4 x[y / x]φ
5 nfsbc1v 2751 . . . 4 x[suc y / x]φ
64, 5nfim 1438 . . 3 x([y / x]φ[suc y / x]φ)
7 sbequ12 1628 . . . 4 (x = y → (φ ↔ [y / x]φ))
8 suceq 4078 . . . . 5 (x = y → suc x = suc y)
98sbceq1d 2738 . . . 4 (x = y → ([suc x / x]φ[suc y / x]φ))
107, 9imbi12d 223 . . 3 (x = y → ((φ[suc x / x]φ) ↔ ([y / x]φ[suc y / x]φ)))
113, 6, 10cbvral 2499 . 2 (x 𝜔 (φ[suc x / x]φ) ↔ y 𝜔 ([y / x]φ[suc y / x]φ))
12 bj-bdfindes.bd . . 3 BOUNDED φ
13 nfsbc1v 2751 . . 3 x[∅ / x]φ
14 sbceq1a 2742 . . . 4 (x = ∅ → (φ[∅ / x]φ))
1514biimprd 147 . . 3 (x = ∅ → ([∅ / x]φφ))
16 sbequ1 1625 . . 3 (x = y → (φ → [y / x]φ))
17 sbceq1a 2742 . . . 4 (x = suc y → (φ[suc y / x]φ))
1817biimprd 147 . . 3 (x = suc y → ([suc y / x]φφ))
1912, 13, 4, 5, 15, 16, 18bj-bdfindis 8403 . 2 (([∅ / x]φ y 𝜔 ([y / x]φ[suc y / x]φ)) → x 𝜔 φ)
2011, 19sylan2b 271 1 (([∅ / x]φ x 𝜔 (φ[suc x / x]φ)) → x 𝜔 φ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1224  [wsb 1619  wral 2276  [wsbc 2733  c0 3193  suc csuc 4041  𝜔com 4229  BOUNDED wbd 8269
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 529  ax-in2 530  ax-io 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-13 1378  ax-14 1379  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996  ax-nul 3847  ax-pr 3908  ax-un 4109  ax-bd0 8270  ax-bdor 8273  ax-bdex 8276  ax-bdeq 8277  ax-bdel 8278  ax-bdsb 8279  ax-bdsep 8341  ax-infvn 8397
This theorem depends on definitions:  df-bi 110  df-tru 1227  df-nf 1324  df-sb 1620  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-ral 2281  df-rex 2282  df-rab 2285  df-v 2529  df-sbc 2734  df-dif 2889  df-un 2891  df-in 2893  df-ss 2900  df-nul 3194  df-sn 3346  df-pr 3347  df-uni 3545  df-int 3580  df-suc 4047  df-iom 4230  df-bdc 8298  df-bj-ind 8384
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator