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Theorem bj-bdfindes 7171
Description: Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 7169 for explanations. From this version, it is easy to prove the bounded version of findes 4253. (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 1402 . . . 4 yφ
2 nfv 1402 . . . 4 y[suc x / x]φ
31, 2nfim 1446 . . 3 y(φ[suc x / x]φ)
4 nfs1v 1797 . . . 4 x[y / x]φ
5 nfsbc1v 2759 . . . 4 x[suc y / x]φ
64, 5nfim 1446 . . 3 x([y / x]φ[suc y / x]φ)
7 sbequ12 1636 . . . 4 (x = y → (φ ↔ [y / x]φ))
8 suceq 4088 . . . . 5 (x = y → suc x = suc y)
98sbceq1d 2746 . . . 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 2507 . 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 2759 . . 3 x[∅ / x]φ
14 sbceq1a 2750 . . . 4 (x = ∅ → (φ[∅ / x]φ))
1514biimprd 147 . . 3 (x = ∅ → ([∅ / x]φφ))
16 sbequ1 1633 . . 3 (x = y → (φ → [y / x]φ))
17 sbceq1a 2750 . . . 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 7169 . 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 1228  [wsb 1627  wral 2284  [wsbc 2741  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-sbc 2742  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: (None)
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