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Theorem bj-findes 15211
Description: Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 15209 for explanations. From this version, it is easy to prove findes 4620. (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 1539 . . . 4 𝑦𝜑
2 nfv 1539 . . . 4 𝑦[suc 𝑥 / 𝑥]𝜑
31, 2nfim 1583 . . 3 𝑦(𝜑[suc 𝑥 / 𝑥]𝜑)
4 nfs1v 1951 . . . 4 𝑥[𝑦 / 𝑥]𝜑
5 nfsbc1v 2996 . . . 4 𝑥[suc 𝑦 / 𝑥]𝜑
64, 5nfim 1583 . . 3 𝑥([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)
7 sbequ12 1782 . . . 4 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8 suceq 4420 . . . . 5 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
98sbceq1d 2982 . . . 4 (𝑥 = 𝑦 → ([suc 𝑥 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
107, 9imbi12d 234 . . 3 (𝑥 = 𝑦 → ((𝜑[suc 𝑥 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)))
113, 6, 10cbvral 2714 . 2 (∀𝑥 ∈ ω (𝜑[suc 𝑥 / 𝑥]𝜑) ↔ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
12 nfsbc1v 2996 . . 3 𝑥[∅ / 𝑥]𝜑
13 sbceq1a 2987 . . . 4 (𝑥 = ∅ → (𝜑[∅ / 𝑥]𝜑))
1413biimprd 158 . . 3 (𝑥 = ∅ → ([∅ / 𝑥]𝜑𝜑))
15 sbequ1 1779 . . 3 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
16 sbceq1a 2987 . . . 4 (𝑥 = suc 𝑦 → (𝜑[suc 𝑦 / 𝑥]𝜑))
1716biimprd 158 . . 3 (𝑥 = suc 𝑦 → ([suc 𝑦 / 𝑥]𝜑𝜑))
1812, 4, 5, 14, 15, 17bj-findis 15209 . 2 (([∅ / 𝑥]𝜑 ∧ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)
1911, 18sylan2b 287 1 (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑[suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  [wsb 1773  wral 2468  [wsbc 2977  c0 3437  suc csuc 4383  ωcom 4607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-nul 4144  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-bd0 15043  ax-bdim 15044  ax-bdan 15045  ax-bdor 15046  ax-bdn 15047  ax-bdal 15048  ax-bdex 15049  ax-bdeq 15050  ax-bdel 15051  ax-bdsb 15052  ax-bdsep 15114  ax-infvn 15171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3613  df-pr 3614  df-uni 3825  df-int 3860  df-suc 4389  df-iom 4608  df-bdc 15071  df-bj-ind 15157
This theorem is referenced by: (None)
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