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Theorem bj-bdfindes 12958
Description: Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 12956 for explanations. From this version, it is easy to prove the bounded version of findes 4485. (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 1491 . . . 4 𝑦𝜑
2 nfv 1491 . . . 4 𝑦[suc 𝑥 / 𝑥]𝜑
31, 2nfim 1534 . . 3 𝑦(𝜑[suc 𝑥 / 𝑥]𝜑)
4 nfs1v 1890 . . . 4 𝑥[𝑦 / 𝑥]𝜑
5 nfsbc1v 2898 . . . 4 𝑥[suc 𝑦 / 𝑥]𝜑
64, 5nfim 1534 . . 3 𝑥([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)
7 sbequ12 1727 . . . 4 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8 suceq 4292 . . . . 5 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
98sbceq1d 2885 . . . 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 2625 . 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 2898 . . 3 𝑥[∅ / 𝑥]𝜑
14 sbceq1a 2889 . . . 4 (𝑥 = ∅ → (𝜑[∅ / 𝑥]𝜑))
1514biimprd 157 . . 3 (𝑥 = ∅ → ([∅ / 𝑥]𝜑𝜑))
16 sbequ1 1724 . . 3 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
17 sbceq1a 2889 . . . 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 12956 . 2 (([∅ / 𝑥]𝜑 ∧ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)
2011, 19sylan2b 283 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 1314  [wsb 1718  wral 2391  [wsbc 2880  c0 3331  suc csuc 4255  ωcom 4472  BOUNDED wbd 12821
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-nul 4022  ax-pr 4099  ax-un 4323  ax-bd0 12822  ax-bdor 12825  ax-bdex 12828  ax-bdeq 12829  ax-bdel 12830  ax-bdsb 12831  ax-bdsep 12893  ax-infvn 12950
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-sn 3501  df-pr 3502  df-uni 3705  df-int 3740  df-suc 4261  df-iom 4473  df-bdc 12850  df-bj-ind 12936
This theorem is referenced by: (None)
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