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

Theorem bj-bdfindes 10074
 Description: Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 10072 for explanations. From this version, it is easy to prove the bounded version of findes 4326. (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 1421 . . . 4 𝑦𝜑
2 nfv 1421 . . . 4 𝑦[suc 𝑥 / 𝑥]𝜑
31, 2nfim 1464 . . 3 𝑦(𝜑[suc 𝑥 / 𝑥]𝜑)
4 nfs1v 1815 . . . 4 𝑥[𝑦 / 𝑥]𝜑
5 nfsbc1v 2782 . . . 4 𝑥[suc 𝑦 / 𝑥]𝜑
64, 5nfim 1464 . . 3 𝑥([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)
7 sbequ12 1654 . . . 4 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8 suceq 4139 . . . . 5 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
98sbceq1d 2769 . . . 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 2529 . 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 2782 . . 3 𝑥[∅ / 𝑥]𝜑
14 sbceq1a 2773 . . . 4 (𝑥 = ∅ → (𝜑[∅ / 𝑥]𝜑))
1514biimprd 147 . . 3 (𝑥 = ∅ → ([∅ / 𝑥]𝜑𝜑))
16 sbequ1 1651 . . 3 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
17 sbceq1a 2773 . . . 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 10072 . 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 1243  [wsb 1645  ∀wral 2306  [wsbc 2764  ∅c0 3224  suc csuc 4102  ωcom 4313  BOUNDED wbd 9932 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883  ax-pr 3944  ax-un 4170  ax-bd0 9933  ax-bdor 9936  ax-bdex 9939  ax-bdeq 9940  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004  ax-infvn 10066 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314  df-bdc 9961  df-bj-ind 10051 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator