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Theorem bj-findes 8361
Description: Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 8359 for explanations. From this version, it is easy to prove findes 4241. (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 1394 . . . 4 yφ
2 nfv 1394 . . . 4 y[suc x / x]φ
31, 2nfim 1437 . . 3 y(φ[suc x / x]φ)
4 nfs1v 1788 . . . 4 x[y / x]φ
5 nfsbc1v 2750 . . . 4 x[suc y / x]φ
64, 5nfim 1437 . . 3 x([y / x]φ[suc y / x]φ)
7 sbequ12 1627 . . . 4 (x = y → (φ ↔ [y / x]φ))
8 suceq 4077 . . . . 5 (x = y → suc x = suc y)
98sbceq1d 2737 . . . 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 2498 . 2 (x 𝜔 (φ[suc x / x]φ) ↔ y 𝜔 ([y / x]φ[suc y / x]φ))
12 nfsbc1v 2750 . . 3 x[∅ / x]φ
13 sbceq1a 2741 . . . 4 (x = ∅ → (φ[∅ / x]φ))
1413biimprd 147 . . 3 (x = ∅ → ([∅ / x]φφ))
15 sbequ1 1624 . . 3 (x = y → (φ → [y / x]φ))
16 sbceq1a 2741 . . . 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 8359 . 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 1223  [wsb 1618  wral 2275  [wsbc 2732  c0 3192  suc csuc 4040  𝜔com 4228
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 529  ax-in2 530  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-13 1377  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-nul 3846  ax-pr 3907  ax-un 4108  ax-setind 4192  ax-bd0 8198  ax-bdim 8199  ax-bdan 8200  ax-bdor 8201  ax-bdn 8202  ax-bdal 8203  ax-bdex 8204  ax-bdeq 8205  ax-bdel 8206  ax-bdsb 8207  ax-bdsep 8269  ax-infvn 8325
This theorem depends on definitions:  df-bi 110  df-tru 1226  df-fal 1229  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-rab 2284  df-v 2528  df-sbc 2733  df-dif 2888  df-un 2890  df-in 2892  df-ss 2899  df-nul 3193  df-sn 3345  df-pr 3346  df-uni 3544  df-int 3579  df-suc 4046  df-iom 4229  df-bdc 8226  df-bj-ind 8312
This theorem is referenced by: (None)
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