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Theorem bj-findes 11364
Description: Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 11362 for explanations. From this version, it is easy to prove findes 4393. (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 1464 . . . 4 𝑦𝜑
2 nfv 1464 . . . 4 𝑦[suc 𝑥 / 𝑥]𝜑
31, 2nfim 1507 . . 3 𝑦(𝜑[suc 𝑥 / 𝑥]𝜑)
4 nfs1v 1860 . . . 4 𝑥[𝑦 / 𝑥]𝜑
5 nfsbc1v 2847 . . . 4 𝑥[suc 𝑦 / 𝑥]𝜑
64, 5nfim 1507 . . 3 𝑥([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)
7 sbequ12 1698 . . . 4 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8 suceq 4205 . . . . 5 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
98sbceq1d 2834 . . . 4 (𝑥 = 𝑦 → ([suc 𝑥 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
107, 9imbi12d 232 . . 3 (𝑥 = 𝑦 → ((𝜑[suc 𝑥 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)))
113, 6, 10cbvral 2582 . 2 (∀𝑥 ∈ ω (𝜑[suc 𝑥 / 𝑥]𝜑) ↔ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑))
12 nfsbc1v 2847 . . 3 𝑥[∅ / 𝑥]𝜑
13 sbceq1a 2838 . . . 4 (𝑥 = ∅ → (𝜑[∅ / 𝑥]𝜑))
1413biimprd 156 . . 3 (𝑥 = ∅ → ([∅ / 𝑥]𝜑𝜑))
15 sbequ1 1695 . . 3 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
16 sbceq1a 2838 . . . 4 (𝑥 = suc 𝑦 → (𝜑[suc 𝑦 / 𝑥]𝜑))
1716biimprd 156 . . 3 (𝑥 = suc 𝑦 → ([suc 𝑦 / 𝑥]𝜑𝜑))
1812, 4, 5, 14, 15, 17bj-findis 11362 . 2 (([∅ / 𝑥]𝜑 ∧ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑[suc 𝑦 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)
1911, 18sylan2b 281 1 (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑[suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1287  [wsb 1689  wral 2355  [wsbc 2829  c0 3275  suc csuc 4168  ωcom 4380
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-nul 3942  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-bd0 11192  ax-bdim 11193  ax-bdan 11194  ax-bdor 11195  ax-bdn 11196  ax-bdal 11197  ax-bdex 11198  ax-bdeq 11199  ax-bdel 11200  ax-bdsb 11201  ax-bdsep 11263  ax-infvn 11324
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-sn 3437  df-pr 3438  df-uni 3639  df-int 3674  df-suc 4174  df-iom 4381  df-bdc 11220  df-bj-ind 11310
This theorem is referenced by: (None)
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