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Theorem bj-findes 12764
Description: Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 12762 for explanations. From this version, it is easy to prove findes 4455. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-findes (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑[suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)

Proof of Theorem bj-findes
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfv 1476 . . . 4 𝑦𝜑
2 nfv 1476 . . . 4 𝑦[suc 𝑥 / 𝑥]𝜑
31, 2nfim 1519 . . 3 𝑦(𝜑[suc 𝑥 / 𝑥]𝜑)
4 nfs1v 1875 . . . 4 𝑥[𝑦 / 𝑥]𝜑
5 nfsbc1v 2880 . . . 4 𝑥[suc 𝑦 / 𝑥]𝜑
64, 5nfim 1519 . . 3 𝑥([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)
7 sbequ12 1712 . . . 4 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8 suceq 4262 . . . . 5 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
98sbceq1d 2867 . . . 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 2608 . 2 (∀𝑥 ∈ ω (𝜑[suc 𝑥 / 𝑥]𝜑) ↔ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
12 nfsbc1v 2880 . . 3 𝑥[∅ / 𝑥]𝜑
13 sbceq1a 2871 . . . 4 (𝑥 = ∅ → (𝜑[∅ / 𝑥]𝜑))
1413biimprd 157 . . 3 (𝑥 = ∅ → ([∅ / 𝑥]𝜑𝜑))
15 sbequ1 1709 . . 3 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
16 sbceq1a 2871 . . . 4 (𝑥 = suc 𝑦 → (𝜑[suc 𝑦 / 𝑥]𝜑))
1716biimprd 157 . . 3 (𝑥 = suc 𝑦 → ([suc 𝑦 / 𝑥]𝜑𝜑))
1812, 4, 5, 14, 15, 17bj-findis 12762 . 2 (([∅ / 𝑥]𝜑 ∧ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)
1911, 18sylan2b 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 1299  [wsb 1703  wral 2375  [wsbc 2862  c0 3310  suc csuc 4225  ωcom 4442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-nul 3994  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-bd0 12592  ax-bdim 12593  ax-bdan 12594  ax-bdor 12595  ax-bdn 12596  ax-bdal 12597  ax-bdex 12598  ax-bdeq 12599  ax-bdel 12600  ax-bdsb 12601  ax-bdsep 12663  ax-infvn 12724
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-sn 3480  df-pr 3481  df-uni 3684  df-int 3719  df-suc 4231  df-iom 4443  df-bdc 12620  df-bj-ind 12710
This theorem is referenced by: (None)
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