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Theorem bj-findes 9441
Description: Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 9439 for explanations. From this version, it is easy to prove findes 4269. (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 1418 . . . 4 yφ
2 nfv 1418 . . . 4 y[suc x / x]φ
31, 2nfim 1461 . . 3 y(φ[suc x / x]φ)
4 nfs1v 1812 . . . 4 x[y / x]φ
5 nfsbc1v 2776 . . . 4 x[suc y / x]φ
64, 5nfim 1461 . . 3 x([y / x]φ[suc y / x]φ)
7 sbequ12 1651 . . . 4 (x = y → (φ ↔ [y / x]φ))
8 suceq 4105 . . . . 5 (x = y → suc x = suc y)
98sbceq1d 2763 . . . 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 2523 . 2 (x 𝜔 (φ[suc x / x]φ) ↔ y 𝜔 ([y / x]φ[suc y / x]φ))
12 nfsbc1v 2776 . . 3 x[∅ / x]φ
13 sbceq1a 2767 . . . 4 (x = ∅ → (φ[∅ / x]φ))
1413biimprd 147 . . 3 (x = ∅ → ([∅ / x]φφ))
15 sbequ1 1648 . . 3 (x = y → (φ → [y / x]φ))
16 sbceq1a 2767 . . . 4 (x = suc y → (φ[suc y / x]φ))
1716biimprd 147 . . 3 (x = suc y → ([suc y / x]φφ))
1812, 4, 5, 14, 15, 17bj-findis 9439 . 2 (([∅ / x]φ y 𝜔 ([y / x]φ[suc y / x]φ)) → x 𝜔 φ)
1911, 18sylan2b 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 1242  [wsb 1642  wral 2300  [wsbc 2758  c0 3218  suc csuc 4068  𝜔com 4256
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-bd0 9268  ax-bdim 9269  ax-bdan 9270  ax-bdor 9271  ax-bdn 9272  ax-bdal 9273  ax-bdex 9274  ax-bdeq 9275  ax-bdel 9276  ax-bdsb 9277  ax-bdsep 9339  ax-infvn 9401
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257  df-bdc 9296  df-bj-ind 9386
This theorem is referenced by: (None)
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