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Theorem bj-bdfindes 13595
Description: Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 13593 for explanations. From this version, it is easy to prove the bounded version of findes 4564. (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 1508 . . . 4 𝑦𝜑
2 nfv 1508 . . . 4 𝑦[suc 𝑥 / 𝑥]𝜑
31, 2nfim 1552 . . 3 𝑦(𝜑[suc 𝑥 / 𝑥]𝜑)
4 nfs1v 1919 . . . 4 𝑥[𝑦 / 𝑥]𝜑
5 nfsbc1v 2955 . . . 4 𝑥[suc 𝑦 / 𝑥]𝜑
64, 5nfim 1552 . . 3 𝑥([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)
7 sbequ12 1751 . . . 4 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8 suceq 4364 . . . . 5 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
98sbceq1d 2942 . . . 4 (𝑥 = 𝑦 → ([suc 𝑥 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
107, 9imbi12d 233 . . 3 (𝑥 = 𝑦 → ((𝜑[suc 𝑥 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)))
113, 6, 10cbvral 2676 . 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 2955 . . 3 𝑥[∅ / 𝑥]𝜑
14 sbceq1a 2946 . . . 4 (𝑥 = ∅ → (𝜑[∅ / 𝑥]𝜑))
1514biimprd 157 . . 3 (𝑥 = ∅ → ([∅ / 𝑥]𝜑𝜑))
16 sbequ1 1748 . . 3 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
17 sbceq1a 2946 . . . 4 (𝑥 = suc 𝑦 → (𝜑[suc 𝑦 / 𝑥]𝜑))
1817biimprd 157 . . 3 (𝑥 = suc 𝑦 → ([suc 𝑦 / 𝑥]𝜑𝜑))
1912, 13, 4, 5, 15, 16, 18bj-bdfindis 13593 . 2 (([∅ / 𝑥]𝜑 ∧ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)
2011, 19sylan2b 285 1 (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑[suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1335  [wsb 1742  wral 2435  [wsbc 2937  c0 3395  suc csuc 4327  ωcom 4551  BOUNDED wbd 13458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-nul 4092  ax-pr 4171  ax-un 4395  ax-bd0 13459  ax-bdor 13462  ax-bdex 13465  ax-bdeq 13466  ax-bdel 13467  ax-bdsb 13468  ax-bdsep 13530  ax-infvn 13587
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-sn 3567  df-pr 3568  df-uni 3775  df-int 3810  df-suc 4333  df-iom 4552  df-bdc 13487  df-bj-ind 13573
This theorem is referenced by: (None)
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